File download and streaming system

ABSTRACT

A method of encoding data operates on an ordered set of input symbols and includes generating redundant symbols from the input symbols, and includes generating output symbols from a combined set of symbols including the input symbols and the redundant symbols, wherein the number of possible output symbols is much larger than the number of the combined set of symbols, wherein at least one output symbol is generated from more than one symbol in the combined set of symbols and from less than all of the symbols in the combined set of symbols. The redundant symbols are generated from an ordered set of input symbols in a deterministic process such that a first set of static symbols calculated using a first input symbol has a low common membership with a second set of static symbols calculated using a second input symbol distinct from the first input symbol.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. patent application Ser. No.11/125,818, filed May 9, 2005 entitled “FILE DOWNLOAD AND STREAMINGSYSTEM,” which claims priority to U.S. Provisional Patent ApplicationNo. 60/569,127, filed May 7, 2004 entitled “FILE DOWNLOAD AND STREAMINGSYSTEM,” which is hereby incorporated by reference, as if set forth infull in this document, for all purposes.

FIELD OF THE INVENTION

The present invention relates to encoding and decoding data incommunications systems and more specifically to communication systemsthat encode and decode data to account for errors and gaps incommunicated data. In embodiments, data is transmitted over broadcastand/or multicast wireless networks to receivers.

BACKGROUND OF THE INVENTION

Transmission of files and streams between a sender and a recipient overa communications channel has been the subject of much literature.Preferably, a recipient desires to receive an exact copy of datatransmitted over a channel by a sender with some level of certainty.Where the channel does not have perfect fidelity (which covers most allphysically realizable systems), one concern is how to deal with datalost or garbled in transmission. Lost data (erasures) are often easierto deal with than corrupted data (errors) because the recipient cannotalways tell when corrupted data is data received in error. Manyerror-correcting codes have been developed to correct for erasuresand/or for errors. Typically, the particular code used is chosen basedon some information about the infidelities of the channel through whichthe data is being transmitted and the nature of the data beingtransmitted. For example, where the channel is known to have longperiods of infidelity, a burst error code might be best suited for thatapplication. Where only short, infrequent errors are expected a simpleparity code might be best.

Data transmission is straightforward when a transmitter and a receiverhave all of the computing power and electrical power needed forcommunications and the channel between the transmitter and receiver isclean enough to allow for relatively error-free communications. Theproblem of data transmission becomes more difficult when the channel isin an adverse environment or the transmitter and/or receiver has limitedcapability.

One solution is the use of forward error correcting (FEC) techniques,wherein data is coded at the transmitter such that a receiver canrecover from transmission erasures and errors. Where feasible, a reversechannel from the receiver to the transmitter allows for the receiver tocommunicate about errors to the transmitter, which can then adjust itstransmission process accordingly. Often, however, a reverse channel isnot available or feasible. For example, where the transmitter istransmitting to a large number of receivers, the transmitter might notbe able to handle reverse channels from all those receivers. As aresult, communication protocols often need to be designed without areverse channel and, as such, the transmitter may have to deal withwidely varying channel conditions without a full view of those channelconditions.

The problem of data transmission between transmitters and receivers ismade more difficult when the receivers need to be low-power, smalldevices that might be portable or mobile and need to receive data athigh bandwidths. For example, a wireless network might be set up todeliver files or streams from a stationary transmitter to a large orindeterminate number of portable or mobile receivers either as abroadcast or multicast where the receivers are constrained in theircomputing power, memory size, available electrical power, antenna size,device size and other design constraints.

In such a system, considerations to be addressed include having littleor no reverse channel, limited memory, limited computing cycles,mobility and timing. Preferably, the design should minimize the amountof transmission time needed to deliver data to potentially a largepopulation of receivers, where individual receivers and might be turnedon and off at unpredictable times, move in and out of range, incurlosses due to link errors, cell changes, congestion in cells forcinglower priority file or stream packets to be temporarily dropped, etc.

In the case of a packet protocol used for data transport, a file, streamor other block of data to be transmitted over a packet network ispartitioned into equal size input symbols and input symbols are placedinto consecutive packets. The “size” of an input symbol can be measuredin bits, whether or not the input symbol is actually broken into a bitstream, where an input symbol has a size of M bits when the input symbolis selected from an alphabet of 2^(M) symbols. In such a packet-basedcommunication system, a packet oriented coding scheme might be suitable.A file transmission is called reliable if it allows the intendedrecipient to recover an exact copy of the original file even in the faceof erasures in the network. A stream transmission is called reliable ifit allows the intended recipient to recover an exact copy of each partof the stream in a timely manner even in the face of erasures in thenetwork. Both file transmission and stream transmission can also besomewhat reliable, in the sense that some parts of the file or streamare not recoverable or for streaming if some parts of the stream are notrecoverable in a timely fashion. Packet loss often occurs becausesporadic congestion causes the buffering mechanism in a router to reachits capacity, forcing it to drop incoming packets. Protection againsterasures during transport has been the subject of much study.

It is known to use chain reaction codes to allow for generation of anarbitrary number of output symbols from the input symbols of a file orstream. This has many uses, including the generation of output symbolsin an information additive way, as opposed to an information duplicativeway, wherein the latter is where a receiver receives additional datathat duplicates data the receiver already knows. Novel techniques forgenerating, using and operating chain reaction codes are shown, forexample, in U.S. Pat. No. 6,307,487 entitled “Information Additive CodeGenerator and Decoder for Communication Systems” issued to Luby (“LubyI″), U.S. Pat. No. 6,320,520 issued to Luby et al. entitled “InformationAdditive Group Code Generator and Decoder for Communication Systems”(hereinafter “Luby II”), and U.S. Published Patent Application No.2003/0058958 published Mar. 27, 2003 and entitled “Multi-Stage CodeGenerator and Decoder for Communication Systems” to Shokrollahi et al.(hereinafter “Shokrollahi”). To the extent permitted, the entiredisclosures of those are herein incorporated herein by reference for allpurposes.

One property of the output symbols produced by a chain reaction encoderis that a receiver is able to recover the original file or block of theoriginal stream as soon as enough output symbols have been received.Specifically, to recover the original K input symbols with a highprobability, the receiver needs approximately K+A output symbols. Theratio A/K is called the “relative reception overhead.” The relativereception overhead depends on the number K of input symbols, and on thereliability of the decoder. For example, in one specific embodiment, andwhere K is equal to 60,000, a relative reception overhead of 5% ensuresthat the decoder successfully decodes the input file or block of thestream with a probability of at least 1-10⁻⁸, and where K is equal to10,000, a relative reception overhead of 15% ensures the same successprobability of the decoder. In one embodiment, the relative receptionoverhead of chain reaction codes can be computed as (13*sqrt(K)+200)/K,where sqrt(K) is the square root of the number of input symbols K. Inthis embodiment the relative reception overhead of chain reaction codestends to be larger for small values of K.

Luby I, Luby II and Shokrollahi provide teachings of systems and methodsthat can be employed in certain embodiments according to the presentinvention. It is to be understood, however, that these systems andmethods are not required of the present invention, and many othervariations, modifications, or alternatives can also be used.

It is also known to use multi-stage chain reaction (“MSCR”) codes, suchas those described in Shokrollahi and developed by Digital Fountain,Inc. under the trade name “Raptor” codes. Multi-stage chain reactioncodes are used, for example, in an encoder that receives input symbolsfrom a source file or source stream, generates intermediate symbolstherefrom and encodes the intermediate symbols using chain reactioncodes. More particularly, a plurality of redundant symbols are generatedfrom an ordered set of input symbols to be transmitted. A plurality ofoutput symbols are generated from a combined set of symbols includingthe input symbols and the redundant symbols, wherein the number ofpossible output symbols is much larger than the number of symbols in thecombined set of symbols, wherein at least one output symbol is generatedfrom more than one symbol in the combined set of symbols and from lessthan all of the symbols in the combined set of symbols, and such thatthe ordered set of input symbols can be regenerated to a desired degreeof accuracy from any predetermined number, N, of the output symbols.

For some applications, other variations of codes might be more suitableor otherwise preferred.

BRIEF SUMMARY OF THE INVENTION

According to one embodiment of the invention, a method of encoding datafor transmission from a source to a destination over a communicationschannel is provided. The method operates on an ordered set of inputsymbols and includes generating a plurality of redundant symbols fromthe input symbols. The method also includes generating a plurality ofoutput symbols from a combined set of symbols including the inputsymbols and the redundant symbols, wherein the number of possible outputsymbols is much larger than the number of symbols in the combined set ofsymbols, wherein at least one output symbol is generated from more thanone symbol in the combined set of symbols and from less than all of thesymbols in the combined set of symbols, and such that the ordered set ofinput symbols can be regenerated to a desired degree of accuracy fromany predetermined number of the output symbols. The plurality ofredundant symbols is generated from an ordered set of input symbols tobe transmitted in a deterministic process such that a first set ofstatic symbols calculated using a first input symbol has a low commonmembership with a second set of static symbols calculated using a secondinput symbol distinct from the first input symbol.

According to still another embodiment of the invention, a system forreceiving data transmitted from a source over a communications channelis provided using similar techniques. The system comprises a receivemodule coupled to a communications channel for receiving output symbolstransmitted over the communications channel, wherein each output symbolis generated from at least one symbol in a combined set of input symbolsand redundant symbols, wherein at least one output symbol is generatedfrom more than one symbol in the combined set and less than all of thesymbols in the combined set, wherein the number of possible outputsymbols is much larger than the number of symbols in the combined set,wherein the input symbols are from an ordered set of input symbols,wherein the redundant symbols are generated from the input symbols andwherein the plurality of redundant symbols is generated from an orderedset of input symbols to be transmitted in a deterministic process suchthat a first set of static symbols calculated using a first input symbolhas a low common membership with a second set of static symbolscalculated using a second input symbol distinct from the first inputsymbol.

According to yet another embodiment of the invention, a computer datasignal embodied in a carrier wave is provided.

Numerous benefits are achieved by way of the present invention. Forexample, in a specific embodiment, the computational expense of encodingdata for transmission over a channel is reduced. In another specificembodiment, the computational expense of decoding such data is reduced.Depending upon the embodiment, one or more of these benefits may beachieved. These and other benefits are provided in more detailthroughout the present specification and more particularly below.

A further understanding of the nature and the advantages of theinventions disclosed herein may be realized by reference to theremaining portions of the specification and the attached drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram of a communications system according to oneembodiment of the present invention.

FIG. 2 is a block diagram an encoder according to one embodiment of thepresent invention.

FIG. 3 is a simplified block diagram of a method of generating redundantsymbols according to one embodiment of the present invention.

FIG. 4 is a simplified block diagram of the basic operation of a staticencoder according to one embodiment of the present invention.

FIG. 5 is a simplified block diagram of a dynamic encoder according toone embodiment of the present invention.

FIG. 6 is a simplified block diagram of a basic operation of a dynamicencoder according to one embodiment of the present invention.

FIG. 7 is a simplified block diagram of a static encoder according toone embodiment of the present invention.

FIG. 8 is a simplified block diagram of the basic operation a staticencoder according to one embodiment of the present invention.

FIG. 9 is a simplified diagram of a method for calculating encodingparameters according to one specific embodiment of a static encoder.

FIG. 10 is a simplified flow diagram of a static encoder according toanother embodiment of the present invention.

FIG. 11 is a simplified block diagram of a decoder according to oneembodiment of the present invention.

FIG. 12 is a simplified flow diagram of an operation of a decoderaccording to one embodiment of the present invention.

FIG. 13 is a simplified flow diagram of an operation of a decoderaccording to another embodiment of the present invention.

FIG. 14 is a simplified flow diagram of an operation of a decoderaccording to yet another embodiment of the present invention.

FIG. 15 is a simplified block diagram of a dynamic decoder according toone embodiment of the present invention.

FIG. 16 is a simplified block diagram of a static decoder according toone embodiment of the present invention.

FIG. 17 illustrates source symbol from sub-symbol mappings.

FIG. 18 illustrates possible settings of file download parameters forvarious file sizes.

FIG. 19 illustrates possible settings of streaming parameters forvarious source block sizes.

FIG. 20 illustrates a form of a matrix that represents a relationshipbetween source and intermediate symbols.

FIG. 21 illustrates a degree distribution for the degree generator.

FIG. 22 illustrates a form of the matrix A that can be used fordecoding.

The detailed description is followed by three appendices: Appendix Acontains example values for systematic indices J(K); Appendix B.1contains example values for table V₀; and Appendix B.2 contains examplevalues for table V₁.

DETAILED DESCRIPTION OF SPECIFIC EMBODIMENTS

In the specific embodiments described herein, a coding scheme denoted as“multi-stage coding” is described, embodiments of which are provided inShokrollahi.

Multi-stage encoding, as described herein, encodes the data in aplurality of stages. Typically, but not always, a first stage adds apredetermined amount of redundancy to the data. A second stage then usesa chain reaction code, or the like, to produce output symbols from theoriginal data and the redundant symbols computed by the first stage ofthe encoding. In one specific embodiment of the present invention, thereceived data is first decoded using a chain reaction decoding process.If that process is not successful in recovering the original datacompletely, a second decoding step can be applied.

In embodiments of multi-stage encoding, redundant symbols are generatedfrom the input file or block of the stream during the first stage ofencoding. In these embodiments, in the second stage of encoding, outputsymbols are generated from the combination of the input file or block ofthe stream and the redundant symbols. In some of these embodiments, theoutput symbols can be generated as needed. In embodiments in which thesecond stage comprises chain reaction encoding, each output symbol canbe generated without regard to how other output symbols are generated.Once generated, these output symbols can then be placed into packets andtransmitted to their destination, with each packet containing one ormore output symbols. Non-packetized transmission techniques can be usedinstead or as well.

As used herein, the term “file” refers to any data that is stored at oneor more sources and is to be delivered as a unit to one or moredestinations. Thus, a document, an image, and a file from a file serveror computer storage device, are all examples of “files” that can bedelivered. Files can be of known size (such as a one megabyte imagestored on a hard disk) or can be of unknown size (such as a file takenfrom the output of a streaming source). Either way, the file is asequence of input symbols, where each input symbol has a position in thefile and a value.

As used herein, the term “stream” refers to any data that is stored orgenerated at one or more sources and is delivered at a specified rate ateach point in time in the order it is generated to one or moredestinations. Streams can be fixed rate or variable rate. Thus, an MPEGvideo stream, AMR audio stream, and a data stream used to control aremote device, are all examples of “streams” that can be delivered. Therate of the stream at each point in time can be known (such as 4megabits per second) or unknown (such as a variable rate stream wherethe rate at each point in time is not known in advance). Either way, thestream is a sequence of input symbols, where each input symbol has aposition in the stream and a value.

Transmission is the process of transmitting data from one or moresenders to one or more recipients through a channel in order to delivera file or stream. A sender is also sometimes referred to as the encoder.If one sender is connected to any number of recipients by a perfectchannel, the received data can be an exact copy of the input file orstream, as all the data will be received correctly. Here, we assume thatthe channel is not perfect, which is the case for most real-worldchannels. Of the many channel imperfections, two imperfections ofinterest are data erasure and data incompleteness (which can be treatedas a special case of data erasure). Data erasure occurs when the channelloses or drops data. Data incompleteness occurs when a recipient doesnot start receiving data until some of the data has already passed itby, the recipient stops receiving data before transmission ends, therecipient chooses to only receive a portion of the transmitted data,and/or the recipient intermittently stops and starts again receivingdata. As an example of data incompleteness, a moving satellite sendermight be transmitting data representing an input file or stream andstart the transmission before a recipient is in range. Once therecipient is in range, data can be received until the satellite movesout of range, at which point the recipient can redirect its satellitedish (during which time it is not receiving data) to start receiving thedata about the same input file or stream being transmitted by anothersatellite that has moved into range. As should be apparent from readingthis description, data incompleteness is a special case of data erasure,since the recipient can treat the data incompleteness (and the recipienthas the same problems) as if the recipient was in range the entire time,but the channel lost all the data up to the point where the recipientstarted receiving data. Also, as is well known in communication systemsdesign, detectable errors can be considered equivalent to erasures bysimply dropping all data blocks or symbols that have detectable errors.

In some communication systems, a recipient receives data generated bymultiple senders, or by one sender using multiple connections. Forexample, to speed up a download, a recipient might simultaneouslyconnect to more than one sender to transmit data concerning the samefile. As another example, in a multicast transmission, multiplemulticast data streams might be transmitted to allow recipients toconnect to one or more of these streams to match the aggregatetransmission rate with the bandwidth of the channel connecting them tothe sender. In all such cases, a concern is to ensure that alltransmitted data is of independent use to a recipient, i.e., that themultiple source data is not redundant among the streams, even when thetransmission rates are vastly different for the different streams, andwhen there are arbitrary patterns of loss.

In general, a communication channel is that which connects the senderand the recipient for data transmission. The communication channel couldbe a real-time channel, where the channel moves data from the sender tothe recipient as the channel gets the data, or the communication channelmight be a storage channel that stores some or all of the data in itstransit from the sender to the recipient. An example of the latter isdisk storage or other storage device. In that example, a program ordevice that generates data can be thought of as the sender, transmittingthe data to a storage device. The recipient is the program or devicethat reads the data from the storage device. The mechanisms that thesender uses to get the data onto the storage device, the storage deviceitself and the mechanisms that the recipient uses to get the data fromthe storage device collectively form the channel. If there is a chancethat those mechanisms or the storage device can lose data, then thatwould be treated as data erasure in the communication channel.

When the sender and recipient are separated by a communication channelin which symbols can be erased, it is preferable not to transmit anexact copy of an input file or stream, but instead to transmit datagenerated from the input file or stream (which could include all orparts of the input file or stream itself) that assists with recovery oferasures. An encoder is a circuit, device, module or code segment thathandles that task. One way of viewing the operation of the encoder isthat the encoder generates output symbols from input symbols, where asequence of input symbol values represent the input file or a block ofthe stream. Each input symbol would thus have a position, in the inputfile or block of the stream, and a value. A decoder is a circuit,device, module or code segment that reconstructs the input symbols fromthe output symbols received by the recipient. In multi-stage coding, theencoder and the decoder are further divided into sub-modules eachperforming a different task.

In embodiments of multi-stage coding systems, the encoder and thedecoder can be further divided into sub-modules, each performing adifferent task. For instance, in some embodiments, the encoder compriseswhat is referred to herein as a static encoder and a dynamic encoder. Asused herein, a “static encoder” is an encoder that generates a number ofredundant symbols from a set of input symbols, wherein the number ofredundant symbols is determined prior to encoding. Examples of staticencoding codes include Reed-Solomon codes, Tornado codes, Hamming codes,Low Density Parity Check (LDPC) codes, etc. The term “static decoder” isused herein to refer to a decoder that can decode data that was encodedby a static encoder.

As used herein, a “dynamic encoder” is an encoder that generates outputsymbols from a set of input symbols, where the number of possible outputsymbols is orders of magnitude larger than the number of input symbols,and where the number of output symbols to be generated need not befixed. One example of a dynamic encoder is a chain reaction encoder,such as the encoders described in Luby I and Luby II. The term “dynamicdecoder” is used herein to refer to a decoder that can decode data thatwas encoded by a dynamic encoder.

Embodiments of multi-stage coding need not be limited to any particulartype of input symbol. Typically, the values for the input symbols areselected from an alphabet of 2^(M) symbols for some positive integer M.In such cases, an input symbol can be represented by a sequence of Mbits of data from the input file or stream. The value of M is oftendetermined based on, for example, the uses of the application, thecommunication channel, and/or the size of the output symbols.Additionally, the size of an output symbol is often determined based onthe application, the channel, and/or the size of the input symbols. Insome cases, the coding process might be simplified if the output symbolvalues and the input symbol values were the same size (i.e.,representable by the same number of bits or selected from the samealphabet). If that is the case, then the input symbol value size islimited when the output symbol value size is limited. For example, itmay be desired to put output symbols in packets of limited size. If somedata about a key associated with the output symbols were to betransmitted in order to recover the key at the receiver, the outputsymbol would preferably be small enough to accommodate, in one packet,the output symbol value and the data about the key.

As an example, if an input file is a multiple megabyte file, the inputfile might be broken into thousands, tens of thousands, or hundreds ofthousands of input symbols with each input symbol encoding thousands,hundreds, or only few bytes. As another example, for a packet-basedInternet channel, a packet with a payload of size of 1024 bytes might beappropriate (a byte is 8 bits). In this example, assuming each packetcontains one output symbol and 8 bytes of auxiliary information, anoutput symbol size of 8128 bits ((1024−8)*8) would be appropriate. Thus,the input symbol size could be chosen as M=(1024−8)*8, or 8128 bits. Asanother example, some satellite systems use the MPEG packet standard,where the payload of each packet comprises 188 bytes. In that example,assuming each packet contains one output symbol and 4 bytes of auxiliaryinformation, an output symbol size of 1472 bits ((188−4)*8), would beappropriate. Thus, the input symbol size could be chosen as M=(188−4)*8,or 1472 bits. In a general-purpose communication system usingmulti-stage coding, the application-specific parameters, such as theinput symbol size (i.e., M, the number of bits encoded by an inputsymbol), might be variables set by the application.

As another example, for a stream that is sent using variable size sourcepackets, the symbol size might be chosen to be rather small so that eachsource packet can be covered with an integral number of input symbolsthat have aggregate size at most slightly larger than the source packet.

Each output symbol has a value. In one preferred embodiment, which weconsider below, each output symbol also has associated therewith anidentifier called its “key.” Preferably, the key of each output symbolcan be easily determined by the recipient to allow the recipient todistinguish one output symbol from other output symbols. Preferably, thekey of an output symbol is distinct from the keys of all other outputsymbols. There are various forms of keying discussed in previous art.For example, Luby I describes various forms of keying that can beemployed in embodiments of the present invention.

Multi-stage coding is particularly useful where there is an expectationof data erasure or where the recipient does not begin and end receptionexactly when a transmission begins and ends. The latter condition isreferred to herein as “data incompleteness.” Regarding erasure events,multi-stage coding shares many of the benefits of chain reaction codingdescribed in Luby I. In particular, multi-stage output symbols areinformation additive, so any suitable number of packets can be used torecover an input file or stream to a desired degree of accuracy. Theseconditions do not adversely affect the communication process whenmulti-stage coding is used, because the output symbols generated withmulti-stage coding are information additive. For example, if a hundredpackets are lost due to a burst of noise causing data erasure, an extrahundred packets can be picked up after the burst to replace the loss ofthe erased packets. If thousands of packets are lost because a receiverdid not tune into a transmitter when it began transmitting, the receivercould just pickup those thousands of packets from any other period oftransmission, or even from another transmitter. With multi-stage coding,a receiver is not constrained to pickup any particular set of packets,so it can receive some packets from one transmitter, switch to anothertransmitter, lose some packets, miss the beginning or end of a giventransmission and still recover an input file or block of a stream. Theability to join and leave a transmission without receiver-transmittercoordination helps to simplify the communication process.

In some embodiments, transmitting a file or stream using multi-stagecoding can include generating, forming or extracting input symbols froman input file or block of a stream, computing redundant symbols,encoding input and redundant symbols into one or more output symbols,where each output symbol is generated based on its key independently ofall other output symbols, and transmitting the output symbols to one ormore recipients over a channel. Additionally, in some embodiments,receiving (and reconstructing) a copy of the input file or block of astream using multi-stage coding can include receiving some set or subsetof output symbols from one of more data streams, and decoding the inputsymbols from the values and keys of the received output symbols.

Suitable FEC erasure codes as described herein can be used to overcomethe above-cited difficulties and would find use in a number of fieldsincluding multimedia broadcasting and multicasting systems and services.An FEC erasure code hereafter referred to as “a multi-stage chainreaction code” has properties that meet many of the current and futurerequirements of such systems and services.

Some basic properties of multi-stage chain reaction codes are that, forany packet loss conditions and for delivery of source files of anyrelevant size or streams of any relevant rate: (a) reception overhead ofeach individual receiver device (“RD”) is minimized; (b) the totaltransmission time needed to deliver source files to any number of RDscan be minimized (c) the quality of the delivered stream to any numberof RDs can be maximized for the number of output symbols sent relativeto the number of input symbols, with suitable selection of transmissionschedules. The RDs might be handheld devices, embedded into a vehicle,portable (i.e., movable but not typically in motion when in use) orfixed to a location.

The amount of working memory needed for decoding is low and can stillprovide the above properties, and the amount of computation needed toencode and decode is minimal. In this document, we provide a simple andeasy to implement description of some variations of multi-stage chainreaction codes.

Multi-stage chain reaction codes are fountain codes, i.e., as manyencoding packets as needed can be generated on-the-fly, each containingunique encoding symbols that are equally useful for recovering a sourcefile or block of a stream. There are many advantages to using fountaincodes versus other types of FEC codes. One advantage is that, regardlessof packet loss conditions and RD availability, fountain codes minimizethe number of encoding packets each RD needs to receive to reconstruct asource file or block of a stream. This is true even under harsh packetloss conditions and when, for example, mobile RDs are onlyintermittently turned-on or available over a long file download session.

Another advantage is the ability to generate exactly as many encodingpackets as needed, making the decision on how many encoding packets togenerate on-the-fly while the transmission is in progress. This can beuseful if for example there is feedback from RDs indicating whether ornot they received enough encoding packets to recover a source file orblock of a stream. When packet loss conditions are less severe thanexpected the transmission can be terminated early. When packet lossconditions are more severe than expected or RDs are unavailable moreoften than expected the transmission can be seamlessly extended.

Another advantage is the ability to inverse multiplex. Inversemultiplexing is when a RD is able to combine received encoding packetsgenerated at independent senders to reconstruct a source file or blockof a stream. One practical use of inverse multiplexing is described inbelow in reference to receiving encoding packets from different senders.

Where future packet loss, RD availability and application conditions arehard to predict, it is important to choose an FEC solution that is asflexible as possible to work well under unpredictable conditionsmulti-stage chain reaction codes provide a degree of flexibilityunmatched by other types of FEC codes.

Aspects of the invention will now be described with reference to thefigures.

System Overview

FIG. 1 is a block diagram of a communications system 100 that usesmulti-stage coding. In communications system 100, an input file 101, oran input stream 105, is provided to an input symbol generator 110. Inputsymbol generator 110 generates a sequence of one or more input symbols(IS(0), IS(1), IS(2), . . . ) from the input file or stream, with eachinput symbol having a value and a position (denoted in FIG. 1 as aparenthesized integer). As explained above, the possible values forinput symbols, i.e., its alphabet, is typically an alphabet of 2^(M)symbols, so that each input symbol codes for M bits of the input file orstream. The value of M is generally determined by the use ofcommunication system 100, but a general purpose system might include asymbol size input for input symbol generator 110 so that M can be variedfrom use to use. The output of input symbol generator 110 is provided toan encoder 115.

Static key generator 130 produces a stream of static keys S₀, S₁, . . .. The number of the static keys generated is generally limited anddepends on the specific embodiment of encoder 115. The generation ofstatic keys will be subsequently described in more detail. Dynamic keygenerator 120 generates a dynamic key for each output symbol to begenerated by the encoder 115. Each dynamic key is generated so that alarge fraction of the dynamic keys for the same input file or block of astream are unique. For example, Luby I describes embodiments of keygenerators that can be used. The outputs of dynamic key generator 120and the static key generator 130 are provided to encoder 115.

From each key I provided by dynamic key generator 120, encoder 115generates an output symbol, with a value B(I), from the input symbolsprovided by the input symbol generator. The operation of encoder 115will be described in more detail below. The value of each output symbolis generated based on its key, on some function of one or more of theinput symbols, and possibly on or more redundant symbols that had beencomputed from the input symbols. The collection of input symbols andredundant symbols that give rise to a specific output symbol is referredto herein as the output symbol's “associated symbols” or just its“associates”. The selection of the function (the “value function”) andthe associates is done according to a process described in more detailbelow. Typically, but not always, M is the same for input symbols andoutput symbols, i.e., they both code for the same number of bits.

In some embodiments, the number K of input symbols is used by theencoder 115 to select the associates. If K is not known in advance, suchas where the input is a streaming file, K can be just an estimate. Thevalue K might also be used by encoder 115 to allocate storage for inputsymbols and any intermediate symbols generated by encoder 115.

Encoder 115 provides output symbols to a transmit module 140. Transmitmodule 140 is also provided the key of each such output symbol from thedynamic key generator 120. Transmit module 140 transmits the outputsymbols, and depending on the keying method used, transmit module 140might also transmit some data about the keys of the transmitted outputsymbols, over a channel 145 to a receive module 150. Channel 145 isassumed to be an erasure channel, but that is not a requirement forproper operation of communication system 100. Modules 140, 145 and 150can be any suitable hardware components, software components, physicalmedia, or any combination thereof, so long as transmit module 140 isadapted to transmit output symbols and any needed data about their keysto channel 145 and receive module 150 is adapted to receive symbols andpotentially some data about their keys from channel 145. The value of K,if used to determine the associates, can be sent over channel 145, or itmay be set ahead of time by agreement of encoder 115 and decoder 155.

As explained above, channel 145 can be a real-time channel, such as apath through the Internet or a broadcast link from a televisiontransmitter to a television recipient or a telephone connection from onepoint to another, or channel 145 can be a storage channel, such as aCD-ROM, disk drive, Web site, or the like. Channel 145 might even be acombination of a real-time channel and a storage channel, such as achannel formed when one person transmits an input file from a personalcomputer to an Internet Service Provider (ISP) over a telephone line,the input file is stored on a Web server and is subsequently transmittedto a recipient over the Internet.

Because channel 145 is assumed to be an erasure channel, communicationssystem 100 does not assume a one-to-one correspondence between theoutput symbols that exit receive module 150 and the output symbols thatgo into transmit module 140. In fact, where channel 145 comprises apacket network, communications system 100 might not even be able toassume that the relative order of any two or more packets is preservedin transit through channel 145. Therefore, the key of the output symbolsis determined using one or more of the keying schemes described above,and not necessarily determined by the order in which the output symbolsexit receive module 150.

Receive module 150 provides the output symbols to a decoder 155, and anydata receive module 150 receives about the keys of these output symbolsis provided to a dynamic key regenerator 160. Dynamic key regenerator160 regenerates the dynamic keys for the received output symbols andprovides these dynamic keys to decoder 155. Static key generator 163regenerates the static keys S₀, S₁, . . . and provides them to decoder155. The static key generator has access to random number generator 135used both during the encoding and the decoding process. This can be inthe form of access to the same physical device if the random numbers aregenerated on such device, or in the form of access to the same algorithmfor the generation of random numbers to achieve identical behavior.Decoder 155 uses the keys provided by dynamic key regenerator 160 andstatic key generator 163 together with the corresponding output symbols,to recover the input symbols (again IS(0), IS(1), IS(2), . . . ).Decoder 155 provides the recovered input symbols to an input filereassembler 165, which generates a copy 170 of input file 101 or inputstream 105.

An Encoder

FIG. 2 is a block diagram of one specific embodiment of encoder 115shown in FIG. 1. Encoder 115 comprises a static encoder 210, a dynamicencoder 220, and a redundancy calculator 230. Static encoder 210receives the following inputs: a) original input symbols IS(0), IS(1), .. . , IS(K−1) provided by the input symbol generator 110 and stored inan input symbol buffer 205; b) the number K of original input symbols;c) static keys S₀, S₁, . . . provided by the static key generator 130;and d) a number R of redundant symbols. Upon receiving these inputsstatic encoder 205 computes R redundant symbols RE(0), RE(1), . . . ,RE(R−1) as will be described below. Typically, but not always, theredundant symbols have the same size as the input symbols. In onespecific embodiment, the redundant symbols generated by static encoder210 are stored in input symbol buffer 205. Input symbol buffer 205 maybe only logical, i.e., the file or block of the stream may be physicallystored in one place and the positions of the input symbols within symbolbuffer 205 could only be renamings of the positions of these symbolswithin the original file or block of the stream.

Dynamic encoder receives the input symbols and the redundant symbols,and generates output symbols as will be described in further detailbelow. In one embodiment in which the redundant symbols are stored inthe input symbol buffer 205, dynamic encoder 220 receives the inputsymbols and redundant symbols from input symbol buffer 205.

Redundancy calculator 230 computes the number R of redundant symbolsfrom the number K of input symbols. This computation is described infurther detail below.

Overview of Static Encoder

The general operation of static encoder 210 is shown with reference toFIGS. 3 and 4. FIG. 3 is a simplified flow diagram illustrating oneembodiment of a method of statically encoding. In a step 305, a variablej, which keeps track of how many redundant symbols have been generated,is set to zero. Then, in a step 310, a first redundant symbol RE(0) iscomputed as a function F₀ of at least some of the input symbols IS(0), .. . , IS(K−1). Then, in a step 315, the variable j is incremented. Next,in a step 320, it is tested whether all of the redundant symbols havebeen generated (i.e., is j greater than R−1?). If yes, then the flowends. Otherwise, the flow proceeds to step 325. In step 325, RE(j) iscomputed as a function F_(j) of the input symbols IS(0), . . . , IS(K−1)and of the previously generated redundant symbols RE(0), . . . ,RE(j−1), where F_(j) need not be a function that depends on every one ofthe input symbols or every one of the redundant symbols. Steps 315, 320,and 325 are repeated until R redundant symbols have been computed.

Referring again to FIGS. 1 and 2, in some embodiments, static encoder210 receives one or more static keys S₀, S₁, . . . from static keygenerator 130. In these embodiments, the static encoder 210 uses thestatic keys to determine some or all of functions F₀, F₁, . . . F_(j-1).For example, static key S₀ can be used to determine function F₀, statickey S₁ can be used to determine function F₁, etc. Or, one or more ofstatic keys S₀, S₁, . . . can be used to determine function F₀, one ormore of static keys S₀, S₁, . . . can be used to determine function F₁,etc. In other embodiments, no static keys are needed, and thus statickey generator 130 is not needed.

Referring now to FIGS. 2 and 3, in some embodiments, the redundantsymbols generated by static encoder 210 can be stored in input symbolbuffer 205. FIG. 4 is a simplified illustration of the operation of oneembodiment of static encoder 210. Particularly, static encoder 210generates redundant symbol RE(j) as a function Fj of input symbolsIS(0), . . . , IS(K−1), RE(0), . . . , RE(j−1), received from inputsymbol buffer 205, and stores it back into input symbol buffer 205. Theexact form of the functions F₀, F₁, . . . , F_(R-1) depends on theparticular application. Typically, but not always, functions F₀, F₁, . .. , F_(R-1) include an exclusive OR of some or all of theircorresponding arguments. As described above, these functions may or maynot actually employ static keys generated by static key generator 130 ofFIG. 1. For example, in one specific embodiment described below, thefirst few functions implement a Hamming code and do not make any use ofthe static keys S₀, S₁, . . . , whereas the remaining functionsimplement a Low-Density Parity-Check code and make explicit use of thestatic keys.

Overview of Multi-Stage Encoder

Referring again to FIG. 2, dynamic encoder 220 receives input symbolsIS(0), . . . , IS(K−1) and the redundant symbols RE(0), . . . , RE(R−1)and a key I for each output symbol it is to generate. The collectioncomprising the original input symbols and the redundant symbols will bereferred to as the collection of “dynamic input symbols” hereafter. FIG.5 is a simplified block diagram of one embodiment of a dynamic encoder,including a weight selector 510, an associator 515, a value functionselector 520 and a calculator 525. As shown in FIG. 5, the K+R dynamicinput symbols are stored in a dynamic symbol buffer 505. In effect,dynamic encoder 500 performs the action illustrated in FIG. 6, namely,to generate an output symbol value B(I) as some value function ofselected input symbols.

FIG. 7 is a simplified block diagram of one specific embodiment of astatic encoder according to the present invention. Static encoder 600comprises a parameter calculator 605, a Hamming encoder 610, and alow-density-parity-check (LDPC) encoder 620. Hamming encoder 610 iscoupled to receive the input symbols IS(0), . . . , IS(K−1) from aninput symbol buffer 625, the number K of input symbols, and theparameter D. In response, Hamming encoder 610 generates D+1 redundantsymbols HA(0), HA(1), . . . , HA(D) according to a Hamming code.

FIG. 8 illustrates the operation of one embodiment of the presentinvention that employs the static encoder shown in FIG. 7.

FIG. 9 is a simplified flow diagram illustrating one embodiment of aparameter calculator, such as parameter calculator 605 of FIG. 7, thatcalculates parameter D and E as described above. First, in a step 705,parameter D is initialized to one. Then, in step 710, it is determinedwhether 2^(D)−D−1 is less than K. If no, then the flow proceeds to step730. If yes, the flow proceeds to step 720, where the parameter D isincremented. Then, the flow proceeds back to step 710. Once D has beendetermined, then, in step 730, the parameter E is calculated as R−D−1.

FIG. 10 is a simplified flow diagram of such an encoder according to oneembodiment of the present invention, which will now be described. First,in step 805, a variable i is initialized to zero. Variable i keeps trackof the number of redundant symbols already generated. In step 810, anumber t is calculated as the smallest odd integer greater than or equalto K/2. In step 815, values P₁, P₂, . . . , P_(t) are generated based onK, t, and a static key S_(i). The values P₁, P₂, . . . , P_(t) indicatethe positions of input symbols that will be used to generate a redundantsymbol. In one particular embodiment, an associator such as associator515 of FIG. 5 is used to generate P₁, P₂, . . . , P_(t). In particular,the value t can be provided as the W(I) input, the value K can beprovided as the K+R input, and the static key S_(i) can be provided asthe key I input. It should be noted that many different values of twould yield similar coding effects, and thus this particular choice isonly an example. In step 820, the value of RE(i) is computed as the XORof the values IS(P₁), IS(P₂), . . . , IS(P_(t)). In step 825, thevariable i is incremented by one to prepare computation of the nextredundant symbol, and in step 830, it is determined whether all theredundant symbols have been computed. If not, then the flow returns tostep 815.

FIG. 11 is a simplified block diagram illustrating one embodiment of adecoder according to the present invention. Decoder 900 can be used, forexample, to implement decoder 155 of FIG. 1.

Decoder 900 comprises a dynamic decoder 905 and a static decoder 910.Input symbols and redundant symbols recovered by dynamic decoder 905 arestored in a reconstruction buffer 915. Upon completion of dynamicdecoding, static decoder 910 attempts to recover any input symbols notrecovered by dynamic decoder 905, if any. In particular, static decoder910 receives input symbols and redundant symbols from reconstructionbuffer 915.

FIG. 12 is a simplified flow diagram illustrating one embodiment of amethod for decoding according to the present invention. In step 1005, Qoutput symbols are received by the decoder. The value of Q can depend onthe number of input symbols and the specific dynamic encoder used. Thevalue of Q can also depend on the desired degree of accuracy to whichthe decoder can recover the input symbols. For example, if it is desiredthat the decoder can recover all of the input symbols with a highprobability, then Q should be chosen to be larger than the number ofinput symbols. Particularly, in some applications, when the number ofinput symbols is large, Q can be less than 3% larger than the number oforiginal input symbols. In other applications, when the number of inputsymbols is small, Q can be at least 10% larger than the number of inputsymbols. Specifically, Q can be chosen as the number K of input symbolsplus a number A, where A is chosen to ensure that the decoder canregenerate all of the input symbols with a high probability.Determination of the number A is described in more detail below. If itis acceptable for the decoder to be unable to decode all of the inputsymbols (either sometimes or always), then Q can be less than K+A, equalto K, or even less than K. Clearly, one aim of an overall coding systemwill often be to decrease the number Q as much as possible, whilemaintaining good probabilistic guarantees on the success of the decodingprocess with respect to the desired degree of accuracy.

In step 1010, dynamic decoder 905 regenerates input symbols andredundant symbols from the Q received output symbols. It is to beunderstood, that steps 1005 and 1010 can be performed substantiallyconcurrently. For example, dynamic decoder 905 can begin regeneratinginput symbols and redundant symbols prior to the decoder receiving Qoutput symbols.

After dynamic decoder 905 has processed Q output symbols, then it isdetermined whether the input symbols have been recovered to a desireddegree of accuracy. The desired degree of accuracy may be, for example,all of the input symbols, or some number, percentage, etc., less thanall of the input symbols. If yes, then the flow ends. If no, then theflow proceeds to step 1020. In step 1020, static decoder 910 attempts torecover any input symbols that dynamic decoder 905 was unable torecover. After static encoder 910 has processed the input symbols andredundant symbols recovered by dynamic encoder 905, then the flow ends.

FIG. 13 is a simplified flow diagram illustrating another embodiment ofa method for decoding according to the present invention. Thisembodiment is similar to that described with respect to FIG. 11, andincludes steps 1005, 1010, 1015, and 1025 in common. But, after step1025, the flow proceeds to step 1030, in which it is determined whetherthe input symbols have been recovered to a desired degree of accuracy.If yes, then the flow ends. If no, then the flow proceeds to step 1035.In step 1035, one or more additional output symbols are received. Then,the flow proceeds back to step 1010, so that dynamic decoder 905 and/orstatic decoder 910 can attempt to recover the remaining unrecoveredinput symbols.

FIG. 14 is a simplified flow diagram illustrating yet another embodimentof a method for decoding according to the present invention. In step1055, output symbols are received by the decoder, and in step 1060,dynamic decoder 905 regenerates input symbols and redundant symbols fromthe received output symbols. Then, in step 1065, it is determinedwhether dynamic decoding should be ended. This determination can bebased on one or more of the number of output symbols processed, thenumber of input symbols recovered, the current rate at which additionalinput symbols are being recovered, the time spent processing outputsymbols, etc.

In step 1065, if it is determined that dynamic decoding is not to bestopped, then the flow proceeds back to step 1055. But, if in step 1065,it is determined to end dynamic decoding, then the flow proceeds to step1070. In step 1070, it is determined whether the input symbols have beenrecovered to a desired degree of accuracy. If yes, then the flow ends.If no, then the flow proceeds to step 1075. In step 1075, static decoder910 attempts to recover any input symbols that dynamic decoder 905 wasunable to recover. After static encoder 910 has processed the inputsymbols and redundant symbols recovered by dynamic encoder 905, the flowends.

FIG. 15 shows one embodiment of dynamic decoder according to the presentinvention. Dynamic decoder 1100 includes similar components as those ofdynamic encoder 500 shown in FIG. 5. Decoder 1100 is similar toembodiments of chain reaction decoders described in Luby I and Luby II.Dynamic decoder 1100 comprises a weight selector 510, an associator 515,a value function selector 520, an output symbol buffer 1105, a reducer1115, a reconstructor 1120 and a reconstruction buffer 1125.

FIG. 16 is a simplified block diagram illustrating one embodiment of astatic decoder. This embodiment can be used when the data is encodedwith a static encoder such as described with reference to FIG. 7. Staticdecoder 1200 comprises a LDPC decoder 1205 and a Hamming decoder 1210.The LDPC decoder 1205 receives input symbols and redundant symbols froma reconstruction buffer 1215, and attempts to reconstruct those symbolsof reconstruction buffer 1215 unrecovered after the decoding step of thedynamic decoder. In some embodiments, reconstruction buffer 1215 isreconstruction buffer 1125 (FIG. 15).

Many variations of LDPC decoders and Hamming decoders are well known tothose skilled in the art, and can be employed in various embodimentsaccording to the present invention. In one specific embodiment, Hammingdecoder is implemented using a Gaussian elimination algorithm. Manyvariations of Gaussian elimination algorithms are well known to thoseskilled in the art, and can be employed in various embodiments accordingto the present invention.

Variations

Multi-stage chain reaction codes as described above are not systematiccodes, i.e., all of the original source symbols of a source block arenot necessarily among the encoding symbols that are sent. However,systematic FEC codes are useful for a file download system or service,and very important for a streaming system or service. As shown in theimplementation below, a modified code can be made to be systematic andstill maintain the fountain code and other described properties.

One reason why it is easy to architect a variety of supplementalservices using multi-stage codes is that it can combine receivedencoding symbols from multiple senders to reconstruct a source file orstream without coordination among the senders. The only requirement isthat the senders use differing sets of keys to generate the encodingsymbols that they send in encoding packets to the code. Ways to achievethis include designating different ranges of the key space to be used byeach such sender, or generating keys randomly at each sender.

As an example of the use of this capability, consider providing asupplemental service to a file download service that allows multi-stagechain reaction codes that did not receive enough encoding packets toreconstruct a source file from the file download session to requestadditional encoding packets to be sent from a make-up sender, e.g., viaa HTTP session. The make-up sender generates encoding symbols from thesource file and sends them, for example using HTTP, and all theseencoding symbols can be combined with those received from the filedownload session to recover the source file. Using this approach allowsdifferent senders to provide incremental source file delivery serviceswithout coordination between the senders, and ensuring that eachindividual receiver need receive only a minimal number of encodingpackets to recover each source file.

Implementations of Various Stages of Multi-Stage Codes

FEC Scheme Definition

A packet using these techniques might be represented with headerinformation such as an FEC Payload ID of four octets comprising a SourceBlock Number (SBN) (16 bit integer identifier for the source block thatthe encoding symbols within the packet relate to) and an Encoding SymbolID (ESI) (16 bit integer identifier for the encoding symbols within thepacket). One suitable interpretation of the Source Block Number andEncoding Symbol Identifier is defined in Sections B below. FEC ObjectTransmission information might comprise the FEC Encoding ID, a TransferLength (F) and the parameters T, Z, N and A defined in below. Theparameters T and Z are 16 bit unsigned integers, N and A are 8 bitunsigned integers.

An FEC encoding scheme for MBMS forward error correction is defined inthe sections below. It defines two different FEC Payload ID formats, onefor FEC source packets and another for FEC repair packets, butvariations for nonsystematic codes are also possible.

The Source FEC payload ID might comprise a Source Block Number (SBN) (16bit integer identifier for the source block that the encoding symbolswithin the packet relate to) and an Encoding Symbol ID (ESI) (16 bitinteger identifier for the encoding symbols within the packet), whilethe Repair FEC Payload ID might comprise a Source Block Number (SBN) (16bit integer identifier for the source block that the repair symbolswithin the packet relate to), an Encoding Symbol ID (ESI) (16 bitinteger identifier for the repair symbols within the packet), and aSource Block Length (SBL) (16 bits, representing the number of sourcesymbols in the source block. The interpretation of the Source BlockNumber, Encoding Symbol Identifier and Source Block Length is definedbelow.

FEC Object Transmission information might comprise the FEC Encoding ID,the maximum source block length, in symbols, and the symbol size, inbytes. The symbol size and maximum source block length might comprise afour octet field of Symbol Size (T) (16 bits representing the size of anencoding symbol, in bytes), and a Maximum Source Block Length (16 bitsrepresenting the maximum length of a source block, in symbols).

The sections below specify the systematic MSCR forward error correctioncode and its application to MBMS and other uses. MSCR is a fountaincode, i.e., as many encoding symbols as needed can be generated by theencoder on-the-fly from the source symbols of a block. The decoder isable to recover the source block from any set of encoding symbols onlyslightly more in number than the number of source symbols. The codedescribed in this document is a systematic code, that is, the originalsource symbols are sent unmodified from sender to receiver, as well as anumber of repair symbols.

B.1 Definitions, Symbols and Abbreviations B.1.1 Definitions

For the purposes of this description, the following terms anddefinitions apply.

Source block: a block of K source symbols which are considered togetherfor MSCR encoding purposes.Source symbol: the smallest unit of data used during the encodingprocess. All source symbols within a source block have the same size.Encoding symbol: a symbol that is included in a data packet. Theencoding symbols comprise the source symbols and the repair symbols.Repair symbols generated from a source block have the same size as thesource symbols of that source block.Systematic code: a code in which the source symbols are included as partof the encoding symbols sent for a source block.Repair symbol: the encoding symbols sent for a source block that are notthe source symbols. The repair symbols are generated based on the sourcesymbols.Intermediate symbols: symbols generated from the source symbols using aninverse encoding process. The repair symbols are then generated directlyfrom the intermediate symbols. The encoding symbols do not include theintermediate symbols, i.e., intermediate symbols are not included indata packets.Symbol: a unit of data. The size, in bytes, of a symbol is known as thesymbol size.Encoding symbol group: a group of encoding symbols that are senttogether, i.e., within the same packet whose relationship to the sourcesymbols can be derived from a single Encoding Symbol ID.Encoding Symbol ID: information that defines the relationship betweenthe symbols of an encoding symbol group and the source symbols.Encoding packet: data packets that contain encoding symbolsSub-block: a source block is sometime broken into sub-blocks, each ofwhich is sufficiently small to be decoded in working memory. For asource block comprising K source symbols, each sub-block comprises Ksub-symbols, each symbol of the source block being composed of onesub-symbol from each sub-block.Sub-symbol: part of a symbol. Each source symbol is composed of as manysub-symbols as there are sub-blocks in the source block.Source packet: data packets that contain source symbols.Repair packet: data packets that contain repair symbols.

B.1.2. Symbols

i, j, x, h, a, b, d, v, m represent positive integers ceil(x) denotesthe smallest positive integer which is greater than or equal to xchoose(i, j) denotes the number of ways j objects can be chosen fromamong i objects without repetition floor(x) denotes the largest positiveinteger which is less than or equal to x i % j denotes i modulo j X{circumflex over ( )} Y denotes, for equal-length bit strings X and Y,the bitwise exclusive-or of X and Y A denote a symbol alignmentparameter. Symbol and sub-symbol sizes are restricted to be multiples ofA. A^(T) denotes the transposed matrix of matrix A A⁻¹ denotes theinverse matrix of matrix A K denotes the number of symbols in a singlesource block K_(MAX) denotes the maximum number of source symbols thatcan be in a single source block. Set to 8192. L denotes the number ofpre-coding symbols for a single source block S denotes the number ofLDPC symbols for a single source block H denotes the number of Halfsymbols for a single source block C denotes an array of intermediatesymbols, C[0], C[1], C[2], . . . , C[L − 1] C′ denotes an array ofsource symbols, C′[0], C′[1], C′[2], . . . , C′[K − 1] X a non-negativeinteger value V₀, V₁ two arrays of 4-byte integers, V₀[0], V₀[1], . . ., V₀[255] and V₁[0], V₁[1], . . . , V₁[255] Rand[X, i, m] apseudo-random number generator Deg[ν] a degree generator LTEnc[K, C, (d,a LT encoding symbol generator a, b)] Trip[K, X] a triple generatorfunction G the number of symbols within an encoding symbol group N thenumber of sub-blocks within a source block T the symbol size in bytes.If the source block is partitioned into sub-blocks, then T = T′ · N. T′the sub-symbol size, in bytes. If the source block is not partitionedinto sub- blocks then T′ is not relevant. F the file size, for filedownload, in bytes I the sub-block size in bytes P for file download,the payload size of each packet, in bytes, that is used in therecommended derivation of the file download transport parameters. Forstreaming, the payload size of each repair packet, in bytes, that isused in the recommended derivation of the streaming transportparameters. Q Q = 65521, i.e., Q is the largest prime smaller than 2¹⁶ Zthe number of source blocks, for file download J(K) the systematic indexassociated with K G denotes any generator matrix I_(S) denotes the S × Sidentity matrix 0_(S×H) denotes the S × H zero matrix

B.1.3 Abbreviations

For the purposes of the present document, the following abbreviationsapply:

ESI Encoding Symbol ID LDPC Low Density Parity Check LT Luby TransformSBN Source Block Number SBL Source Block Length (in units of symbols)

B.2. Overview

The MSCR forward error correction code can be applied to both MBMS filedelivery and MBMS streaming applications. MSCR code aspects which arespecific to each of these applications are discussed in Sections B.3 andB.4 of this document.

A component of the systematic MSCR code is the basic encoder describedin Section B.5. First, it is described how to derive values for a set ofintermediate symbols from the original source symbols such thatknowledge of the intermediate symbols is sufficient to reconstruct thesource symbols. Secondly, the encoder produces repair symbols which areeach the exclusive OR of a number of the intermediate symbols. Theencoding symbols are the combination of the source and repair symbols.The repair symbols are produced in such a way that the intermediatesymbols and therefore also the source symbols can be recovered from anysufficiently large set of encoding symbols.

This document defines the systematic MSCR code encoder. A number ofpossible decoding algorithms are possible. An efficient decodingalgorithm is provided in Section B.6.

The construction of the intermediate and repair symbols is based in parton a pseudo-random number generator described in Section B.5. Thisgenerator is based on a fixed set of 512 random numbers that areavailable to both sender and receiver. An example set of numbers arethose provided in Appendix B.1.

Finally, the construction of the intermediate symbols from the sourcesymbols is governed by a “systematic index”. An example set of valuesfor the systematic index is shown in Appendix A for source block sizesfrom 4 source symbols to K_(MAX)=8192 source symbols.

B.3. File Download B.3.1. Source Block Construction B.3.1.1. General

In order to apply the MSCR encoder to a source file, the file may bebroken into Z≧1 blocks, known as source blocks. The MSCR encoder isapplied independently to each source block. Each source block isidentified by a unique integer Source Block Number (SBN), where thefirst source block has SBN zero, the second has SBN one, etc. Eachsource block is divided into a number, K, of source symbols of size Tbytes each. Each source symbol is identified by a unique integerEncoding Symbol Identifier (ESI), where the first source symbol of asource block has ESI zero, the second has ESI one, etc.

Each source block with K source symbols is divided into N≧1 sub-blocks,which are small enough to be decoded in the working memory. Eachsub-block is divided into K sub-symbols of size T′.

Note that the value of K is not necessarily the same for each sourceblock of a file and the value of T′ may not necessarily be the same foreach sub-block of a source block. However, the symbol size T is the samefor all source blocks of a file and the number of symbols, K is the samefor every sub-block of a source block. Exact partitioning of the fileinto source blocks and sub-blocks is described in B.3.1.2 below.

FIG. 17 shows an example source block placed into a two dimensionalarray, where each entry is a T′-byte sub-symbol, each row is a sub-blockand each column is a source symbol. In this example, the value of T′ isthe same for every sub-block. The number shown in each sub-symbol entryindicates their original order within the source block. For example, thesub-symbol numbered K contains bytes T′·K through T′·(K+1)−1 of thesource block. Then, source symbol i is the concatenation of the ithsub-symbol from each of the sub-blocks, which corresponds to thesub-symbols of the source block numbered i, K+i, 2·K+i, . . . ,(N−1)·K+i.

B.3.1.2 Source Block and Sub-Block Partitioning

The construction of source blocks and sub-blocks is determined based onfive input parameters, F, A, T, Z and N and a function Partition[ ]. Thefive input parameters are defined as follows:

F the size of the file, in bytes A a symbol alignment parameter, inbytes T the symbol size, in bytes, which must be a multiple of A Z thenumber of source blocks N the number of sub-blocks in each source block

These parameters might be set so that ceil(ceil(F/T)/Z)≦K_(MAX).Recommendations for derivation of these parameters are provided inSection B.3.4.

The function Partition[ ] takes a pair of integers (I, J) as input andderives four integers (I_(L), I_(S), J_(L), J_(S)) as output.Specifically, the value of Partition[I, J] is a sequence of fourintegers (I_(L), I_(S), J_(L), J_(S)), where I_(L)=ceil(I/J),I_(S)=floor(I/J), J_(L)=I−I_(S)·J and J_(S)=J−J_(L). Partition[ ]derives parameters for partitioning a block of size I into Japproximately equal sized blocks. Specifically, J_(L) blocks of lengthI_(L) and J_(S) blocks of length I_(S).

The source file might be partitioned into source blocks and sub-blocksas follows:

Let, K_(t) = ceil(F/T) (K_(L), K_(S), Z_(L), Z_(S)) = Partition[K_(t),Z]   (T_(L), T_(S), N_(L), N_(S)) = Partition[T/A, N]

Then, the file might be partitioned into Z=Z_(L)+Z_(S) contiguous sourceblocks, the first Z_(L) source blocks each having length K_(L)·T bytesand the remaining Z_(S) source blocks each having K_(S)·T bytes.

If K_(t)·T>F then for encoding purposes, the last symbol might be paddedat the end with K_(t)·T−F zero bytes.

Next, each source block might be divided into N=N_(L)+N_(S) contiguoussub-blocks, the first N_(L) sub-blocks each comprising K contiguoussub-symbols of size of T_(L)·A and the remaining N_(S) sub-blocks eachcomprising K contiguous sub-symbols of size of T_(S)·A. The symbolalignment parameter A ensures that sub-symbols are always a multiple ofA bytes.

Finally, the mth symbol of a source block comprises the concatenation ofthe mth sub-symbol from each of the N sub-blocks.

B.3.2. Encoding Packet Construction B.3.2.1. General

Each encoding packet contains the following information:

Source Block Number (SBN) Encoding Symbol ID (ESI)

encoding symbol(s)

Each source block is encoded independently of the others. Source blocksare numbered consecutively from zero.

Encoding Symbol JD values from 0 to K−1 identify the source symbols.Encoding Symbol IDs from K onwards identify repair symbols.

B.3.2.2 Encoding Packet Construction

Each encoding packet preferably either consists entirely of sourcesymbols (source packet) or entirely of repair symbols (repair packet). Apacket may contain any number of symbols from the same source block. Inthe case that the last symbol in the packet includes padding bytes addedfor FEC encoding purposes then these bytes need not be included in thepacket. Otherwise, only whole symbols might be included.

The Encoding Symbol ID, X, carried in each source packet is the EncodingSymbol ID of the first source symbol carried in that packet. Thesubsequent source symbols in the packet have Encoding Symbol IDs, X+1 toX+G−1, in sequential order, where G is the number of symbols in thepacket.

Similarly, the Encoding Symbol ID, X, placed into a repair packet is theEncoding Symbol ID of the first repair symbol in the repair packet andthe subsequent repair symbols in the packet have Encoding Symbol IDs X+1to X+G−1 in sequential order, where G is the number of symbols in thepacket.

Note that it is not necessary for the receiver to know the total numberof repair packets. The G repair symbol triples (d[0], a[0], b[0]), . . ., (d[G−1], a[G−1], b[G−1]) for the repair symbols placed into a repairpacket with ESI X are computed using the Triple generator defined inB.5.3.4 as follows:

For each i=0, . . . , G−1

(d[i],a[i],b[i])=Trip[K,X+i]

The G repair symbols to be placed in repair packet with ESI X arecalculated based on the repair symbol triples as described in SectionB.5.3 using the intermediate symbols C and the LT encoder LTenc[K, C,(d[i], a[i], b[i])].

B.3.3. Transport

This section describes the information exchange between the MSCRencoder/decoder and any transport protocol making use of MSCR forwarderror correction for file delivery.

The MSCR encoder and decoder for file delivery require the followinginformation from the transport protocol: the file size, F, in bytes, thesymbol alignment parameter, A, the symbol size, T, in bytes, which is amultiple of A, the number of source blocks, Z, the number of sub-blocksin each source block, N. The MSCR encoder for file delivery additionallyrequires the file to be encoded, F bytes.

The MSCR encoder supplies the transport protocol with encoding packetinformation comprising, for each packet, the SBN, the ESI and theencoding symbol(s). The transport protocol might communicate thisinformation transparently to the MSCR decoder.

B.3.4. Recommended Parameters (Informative) B.3.4.1 Parameter DerivationAlgorithm

This section provides recommendations for the derivation of the fourtransport parameters, A, T, Z and N. This recommendation is based on thefollowing input parameters:

F the file size, in bytesW a target on the sub-block size, in bytesP the maximum packet payload size, in bytes, which is assumed to be amultiple of AA the symbol alignment factor, in bytesK_(MAX) the maximum number of source symbols per source block.K_(MIN) a minimum target on the number of symbols per source blockG_(MAX) a maximum target number of symbols per packet

Based on the above inputs, the transport parameters T, Z and N arecalculated as follows:

Let, G = min{ceil(P · K_(MIN)/F), P/A, G_(MAX)} - the approximate numberof symbols per packet T = floor(P/(A · G)) · A K_(t) = ceil(F/T) - thetotal number of symbols in the file   Z = ceil(K_(t)/K_(MAX))   N =min{ceil(ceil(K_(t)/Z) · T/W), T/A}

The values of G and N derived above should be considered as lowerbounds. It may be advantageous to increase these values, for example tothe nearest power of two. In particular, the above algorithm does notguarantee that the symbol size, T, divides the maximum packet size, P,and so it may not be possible to use the packets of size exactly P. If,instead, G is chosen to be a value which divides P/A, then the symbolsize, T, will be a divisor of P and packets of size P can be used.

Recommended settings for the input parameters, W, A, K_(MIN) and G_(MAX)are as follows:

W=256 KB A=4 K_(MIN)=1024 G_(MAX)=10 B.3.4.2 Examples

The above algorithm leads to transport parameters as shown in FIG. 18,assuming the recommended values for W, A, K_(MIN) and G_(MAX) and P=512.

B.4. Streaming B.4.1. Source Block Construction

A source block is constructed by the transport protocol, for example asdefined in this document, making use of the Systematic MSCR ForwardError Correction code. The symbol size, T, to be used for source blockconstruction and the repair symbol construction are provided by thetransport protocol. The parameter T might be set so that the number ofsource symbols in any source block is at most K_(MAX).

Recommended parameters are presented in section B.4.4.

B.4.2. Encoding Packet Construction

As described in B.4.3., each repair packet contains the SBN, ESI, SBLand repair symbol(s). The number of repair symbols contained within arepair packet is computed from the packet length. The ESI values placedinto the repair packets and the repair symbol triples used to generatethe repair symbols are computed as described in Section B.3.2.2.

B.4.3. Transport

This section describes the information exchange between the MSCRencoder/decoder and any transport protocol making use of MSCR forwarderror correction for streaming. The MSCR encoder for streaming might usethe following information from the transport protocol for each sourceblock: the symbol size, T, in bytes, the number of symbols in the sourceblock, K, the Source Block Number (SBN) and the source symbols to beencoded, K·T bytes. The MSCR encoder supplies the transport protocolwith encoding packet information comprising, for each repair packet, theSBN, the ESI, the SBL and the repair symbol(s). The transport protocolmight communicate this information transparently to the MSCR decoder.

B.4.4. Recommended Parameters B.4.4.1 Parameter Derivation Algorithm

This section provides recommendations for the derivation of thetransport parameter T. This recommendation is based on the followinginput parameters:

B the maximum source block size, in bytes P the maximum repair packetpayload size, in bytes, which is a multiple of A A the symbol alignmentfactor, in bytes K_(MAX) the maximum number of source symbols per sourceblock. K_(MIN) a minimum target on the number of symbols per sourceblock G_(MAX) a maximum target number of symbols per repair packet

A requirement on these inputs is that ceil(B/P)≦K_(MAX). Based on theabove inputs, the transport parameter T is calculated as follows:

Let G=min{ceil(P·K_(MIN)/B), P/A, G_(MAX)}—the approximate number ofsymbols per packet

T=floor(P/(A·G))·A

The value of T derived above should be considered as a guide to theactual value of T used. It may be advantageous to ensure that T dividesinto P, or it may be advantageous to set the value of T smaller tominimize wastage when full size repair symbols are used to recoverpartial source symbols at the end of lost source packets (as long as themaximum number of source symbols in a source block does not exceedK_(MAX)). Furthermore, the choice of T may depend on the source packetsize distribution, e.g., if all source packets are the same size then itis advantageous to choose T so that the actual payload size of a repairpacket P′, where P′ is a multiple of T, is equal to (or as few bytes aspossible larger than) the number of bytes each source packet occupies inthe source block.

Recommended settings for the input parameters, A, K_(MIN) and G_(MAX)are as follows:

A=4 K_(MIN)=1024 G_(MAX)=10 B.4.4.2Examples

The above algorithm leads to transport parameters as shown in FIG. 19,assuming the recommended values for A, K_(MIN) and G_(MAX) and P=512.

B.5. Systematic MSCR Encoder B.5.1. Encoding Overview

The systematic MSCR encoder is used to generate repair symbols from asource block that comprises K source symbols.

Symbols are the fundamental data units of the encoding and decodingprocess. For each source block (sub-block) all symbols (sub-symbols) arethe same size. The atomic operation performed on symbols (sub-symbols)for both encoding and decoding is the exclusive- or operation.

Let C′[0], . . . , C′[K−1] denote the K source symbols.Let C[0], . . . , C[L−1] denote L intermediate symbols.

The first step of encoding is to generate a number, L>K, of intermediatesymbols from the K source symbols. In this step, K source triples (d[0],a[0], b[0]), . . . , (d[K−1], a[K−1], b[K−1]) are generated using theTrip[ ] generator as described in Section B.5.4.4. The K source triplesare associated with the K source symbols and are then used to determinethe L intermediate symbols C[0], . . . , C[L−1] from the source symbolsusing an inverse encoding process. This process can be can be realizedby a MSCR decoding process.

Certain “pre-coding relationships” must hold within the L intermediatesymbols. Section B.5.2 describes these relationships and how theintermediate symbols are generated from the source symbols.

Once the intermediate symbols have been generated, repair symbols areproduced and one or more repair symbols are placed as a group into asingle data packet. Each repair symbol group is associated with anEncoding Symbol ID (ESI) and a number, G, of encoding symbols. The ESIis used to generate a triple of three integers, (d, a, b) for eachrepair symbol again using the Trip[ ] generator as described in SectionB.5.4.4. This is done as described in Sections B.3 and B.4 using thegenerators described in Section B.5.4. Then, each (d,a,b)-triple is usedto generate the corresponding repair symbol from the intermediatesymbols using the LTEnc[K, C[0], . . . , C[L−1], (d,a,b)] generatordescribed in Section B.5.4.3.

B.5.2. First Encoding Step: Intermediate Symbol Generation B.5.2.1General

The first encoding step is a pre-coding step to generate the Lintermediate symbols C[0], . . . , C[L−1] from the source symbols C′[0],. . . , C′[K−1]. The intermediate symbols are uniquely defined by twosets of constraints:

1. The intermediate symbols are related to the source symbols by a setof source symbol triples. The generation of the source symbol triples isdefined in Section B.5.2.2 using the Trip[ ] generator as described inSection B.5.4.4.2. A set of pre-coding relationships hold within the intermediatesymbols themselves. These are defined in Section B.5.2.3.

The generation of the L intermediate symbols is then defined in Section5.2.4.

B.5.2.2 Source Symbol Triples

Each of the K source symbols is associated with a triple (d[i], a[i],b[i]) for 0≦i<K. The source symbol triples are determined using theTriple generator defined in Section B.5.4.4 as:

For each i, 0≦i<K

(d[i],a[i],b[i])=Trip[K,i]

B.5.2.3 Pre-Coding Relationships

The pre-coding relationships amongst the L intermediate symbols aredefined by expressing the last L-K intermediate symbols in terms of thefirst K intermediate symbols.

The last L-K intermediate symbols C[K], . . . , C[L−1] comprise S LDPCsymbols and H Half symbols The values of S and H are determined from Kas described below. Then L=K+S+H.

Let X be the smallest positive integer such that X · (X − 1) = 2 · K. Sbe the smallest prime integer such that S ≧ ceil(0.01 · K) + X H be thesmallest integer such that choose(H, ceil(H/2)) ≧ K + S H′ = ceil(H/2)L= K + S + H C[0], ..., C[K − 1] denote the first K intermediate symbolsC[K], ..., C[K + S − 1] denote the S LDPC symbols, initialised to zeroC[K + S], ..., C[L − 1] denote the H Half symbols, initialised to zero

The S LDPC symbols are defined to be the values of C[K], . . . ,C[K+S−1] at the end of the following process:

For i = 0, . . . ,K−1 do a = 1 + (floor(i/S) % (S−1)) b = i % S C[K + b]= C[K + b] {circumflex over ( )}C[i] b = (b + a) % S C[K + b] = C[K + b]{circumflex over ( )}C[i] b = (b + a) % S C[K + b] = C[K + b]{circumflex over ( )}C[i]

The H Half symbols are defined as follows:

Let g[i] = i {circumflex over ( )}(floor(i/2)) for all positive integersi Note: g[i] is the Gray sequence, in which each element differs fromthe previous one in a single bit position g[j, k] denote the j^(th)element, j = 0, 1, 2, ..., of the subsequence of g[i] whose elementshave exactly k non-zero bits in their binary representation

Then, the Half symbols are defined as the values of C[K+S], . . . ,C[L−1] after the following process:

For h = 0,...,H−1 do For j = 0,...,K+S−1 do If bit h of g[j,H′] is equalto 1 then C[h+K+S] = C[h+K+S] {circumflex over ( )} C[j].

B.5.2.4 Intermediate Symbols B.5.2.4.1 Definition

Given the K source symbols C′[0], C′[1], . . . , C′[K−1] the Lintermediate symbols C[0], C[1], . . . , C[L−1] are the uniquely definedsymbol values that satisfy the following conditions:

1. The K source symbols C′[0], C′[1], . . . , C′[K−1] satisfy the Kconstraints

C′[i]≡LTEnc[K,(C[0], . . . , C[L−1]), (d[i],a[i],b[i])], for all i,0≦i<K.

2. The L intermediate symbols C[0], C[1], . . . , C[L−1] satisfy thepre-coding relationships defined in B.5.2.3.

B.5.2.4.2 Calculation of Intermediate Symbols

This subsection describes a possible method for calculation of the Lintermediate symbols C[0], C[1], . . . , C[L−1] satisfying theconstraints in B.5.2.4.1

The generator matrix G for a code which generates N output symbols fromK input symbols is an N×K matrix over GF(2), where each row correspondsto one of the output symbols and each column to one of the input symbolsand where the i^(th) output symbol is equal to the sum of those inputsymbols whose column contains a non-zero entry in row i.

Then, the L intermediate symbols can be calculated as follows:

Let

C denote the column vector of the L intermediate symbols, C[0], C[1], .. . , C[L−1].D denote the column vector comprising S+H zero symbols followed by the Ksource symbols C′[0], C′[1], . . . , C′[K−1]Then the above constraints define an L×L matrix over GF(2), A, suchthat:

A·C=D

The matrix A can be constructed as follows:

Let: G_(LDPC) be the S × K generator matrix of the LDPC symbols. So,G_(LDPC) · (C[0], ..., C[K − 1])^(T) = (C[K], ..., C[K + S − 1])^(T)G_(Half) be the H × (K + S) generator matrix of the Half symbols, So,G_(Half) · (C[0], ..., C[S + K − 1])^(T) = (C[K + S], ..., C[K + S + H −1])^(T) I_(S) be the S × S identity matrix I_(H) be the H × H identitymatrix 0_(S×H) be the S × H zero matrix G_(LT) be the K × L generatormatrix of the encoding symbols generated by the LT Encoder. So, G_(LT) ·(C[0], ..., C[L − 1])^(T) = (C′[0], C′[1], ..., C′[K − 1])^(T) i.e.G_(LTi,j) = 1 if and only if C[i] is included in the symbols which areXORed to produce LTEnc[K, (C[0], ..., C[L − 1]), (d[i], a[i], b[i])].Then: The first S rows of A are equal to G_(LDPC)|I_(S)|Z_(S×H). Thenext H rows of A are equal to G_(Half)|I_(H). The remaining K rows of Aare equal to G_(LT).

The matrix A is depicted in FIG. 20. The intermediate symbols can thenbe calculated as:

C=A ⁻¹ ·D

The source triples are generated such that for any K matrix A has fullrank and is therefore invertible. This calculation can be realized byapplying a MSCR decoding process to the K source symbols C′[0], C′[1], .. . , C′[K−1] to produce the L intermediate symbols C[0], C[1], . . . ,C[L−1].

To efficiently generate the intermediate symbols from the sourcesymbols, it is recommended that an efficient decoder implementation suchas that described in Section B.6 be used. The source symbol triples aredesigned to facilitate efficient decoding of the source symbols usingthat algorithm.

B.5.3. Second Encoding Step: LT Encoding

In the second encoding step, the repair symbol with ESI X is generatedby applying the generator LTEnc[K, (C[0], C[1], . . . , C[L−1]), (d, a,b)] defined in Section B.5.4 to the L intermediate symbols C[0], C[1], .. . , C[L−1] using the triple (d, a, b)=Trip[K,X] generated according toSections B.3.2.2 and B.4.2.

B.5.4. Generators B.5.4.1 Random Generator

The random number generator Rand[X, i, m] is defined as follows, where Xis anon-negative integer, i is a non-negative integer and m is apositive integer and the value produced is an integer between 0 and m−1.Let V₀ and V₁ be arrays of 256 entries each, where each entry is a4-byte unsigned integer. These arrays are provided in Section B.7.

Then,

Rand[X,i,m]=(V ₀[(X+i)%256]̂V ₁[(floor(X/256)+i)%256])%m

B.5.4.2 Degree Generator

The degree generator Deg[v] is defined as follows, where v is an integerthat is at least 0 and less than 2²⁰=1048576.

In FIG. 21, find the index j such that f[j−1]≦v<f[j]

Deg[v]=d[j]

B.5.4.3 LT Encoding Symbol Generator

The encoding symbol generator LTEnc[K, (C[0], C[1], . . . , C[L−1]), (d,a, b)] takes the following inputs:

K is the number of source symbols (or sub-symbols) for the source block(sub-block). Let L be derived from K as described in Section B.5.2, andlet L′ be the smallest prime integer greater than or equal to L.(C[0], C[1], . . . , C[L−1]) is the array of L intermediate symbols(sub-symbols) generated as described in Section B.5.2(d, a, b) is a source triple determined using the Triple generatordefined in Section B.5.3.4, whereby d is an integer denoting an encodingsymbol degree, a is an integer between 1 and L′−1 inclusive and b is aninteger between 0 and L′−1 inclusive.

The encoding symbol generator produces a single encoding symbol asoutput, according to the following algorithm:

While (b ≧ L) do b = (b + a) % L’ LTEnc[K, (C[0], C[1],..., C[L−1]), (d,a, b)] = C[b]. For j = 1,...,min(d−1,L−1) do b = (b + a) % L’ While (b ≧L) do b = (b + a) % L’ LTEnc[K, (C[0], C[1],..., C[L−1]), (d, a, b)] =LTEnc[K, (C[0], C[1],..., C[L−1]), (d, a, b)] {circumflex over ( )} C[b]

B.5.4.4Triple Generator

The triple generator Trip[K,X] takes the following inputs:

K The number of source symbolsX An encoding symbol ID

Let

L be determined from K as described in Section B.5.2L′ be the smallest prime that is greater than or equal to LQ=65521, the largest prime smaller than 2¹⁶.J(K) be the systematic index associated with K, as defined in Appendix A

The output of the triple generator is a triples, (d, a, b) determined asfollows:

1. A=(53591+J(K)·997) % Q 2. B=10267·(J(K)+1) % Q 3. Y=(B+X·A) % Q

4. v=Rand[Y, 0, 2²⁰]5. d=Deg[v]6. a=1+Rand[Y, 1, L′−1]7. b=Rand[Y, 2, L′]

B.6 FEC Decoder Implementations B.6.1 General

This section describes an efficient decoding algorithm for the MSCRcodes described in this specification. Note that each received encodingsymbol can be considered as the value of an equation amongst theintermediate symbols. From these simultaneous equations, and the knownpre-coding relationships amongst the intermediate symbols, any algorithmfor solving simultaneous equations can successfully decode theintermediate symbols and hence the source symbols. However, thealgorithm chosen has a major effect on the computational efficiency ofthe decoding.

B.6.2 Decoding a Source Block B.6.2.1 General

It is assumed that the decoder knows the structure of the source blockit is to decode, including the symbol size, T, and the number K ofsymbols in the source block.

From the algorithms described in Sections B.5, the MSCR decoder cancalculate the total number L=K+S+H of pre-coding symbols and determinehow they were generated from the source block to be decoded. In thisdescription it is assumed that the received encoding symbols for thesource block to be decoded are passed to the decoder. Furthermore, foreach such encoding symbol it is assumed that the number and set ofintermediate symbols whose exclusive- or is equal to the encoding symbolis passed to the decoder. In the case of source symbols, the sourcesymbol triples described in Section B.5.2.2 indicate the number and setof intermediate symbols which sum to give each source symbol.

Let N≧K be the number of received encoding symbols for a source blockand let M=S+H+N. The following M by L bit matrix A can be derived fromthe information passed to the decoder for the source block to bedecoded. Let C be the column vector of the L intermediate symbols, andlet D be the column vector of M symbols with values known to thereceiver, where the first S+H of the M symbols are zero-valued symbolsthat correspond to LDPC and Half symbols (these are check symbols forthe LDPC and Half symbols, and not the LDPC and Half symbolsthemselves), and the remaining N of the M symbols are the receivedencoding symbols for the source block. Then, A is the bit matrix thatsatisfies A·C=D, where here · denotes matrix multiplication over GF[2].In particular, A[i,j]=1 if the intermediate symbol corresponding toindex j is exclusive-ORed into the LDPC, Half or encoding symbolcorresponding to index i in the encoding, or if index i corresponds to aLDPC or Half symbol and index j corresponds to the same LDPC or Halfsymbol. For all other i and j, A[i,j]=0.

Decoding a source block is equivalent to decoding C from known A and D.It is clear that C can be decoded if and only if the rank of A overGF[2] is L. Once C has been decoded, missing source symbols can beobtained by using the source symbol triples to determine the number andset of intermediate symbols which are exclusive-ORed to obtain eachmissing source symbol.

The first step in decoding C is to form a decoding schedule. In thisstep A is converted, using Gaussian elimination (using row operationsand row and column reorderings) and after discarding M-L rows, into theL by L identity matrix. The decoding schedule comprises the sequence ofrow operations and row and column re-orderings during the Gaussianelimination process, and only depends on A and not on D. The decoding ofC from D can take place concurrently with the forming of the decodingschedule, or the decoding can take place afterwards based on thedecoding schedule.

The correspondence between the decoding schedule and the decoding of Cis as follows. Let c[0]=0, c[1]=1 . . . , c[L−1]=L−1 and d[0]=0, d[1]=1. . . , d[M−1]=M−1 initially.

-   -   Each time row i of A is exclusive-ORed into row i′ in the        decoding schedule then in the decoding process symbol D[d[i]] is        exclusive-ORed into symbol D[d[i′]].    -   Each time row i is exchanged with row i′ in the decoding        schedule then in the decoding process the value of d[i] is        exchanged with the value of d[i′].    -   Each time column j is exchanged with column j′ in the decoding        schedule then in the decoding process the value of c[j] is        exchanged with the value of c[j′].

From this correspondence it is clear that the total number ofexclusive-ORs of symbols in the decoding of the source block is thenumber of row operations (not exchanges) in the Gaussian elimination.Since A is the L by L identity matrix after the Gaussian elimination andafter discarding the last M-L rows, it is clear at the end of successfuldecoding that the L symbols D[d[0]], D[d[1]], . . . , D[d[L−1]] are thevalues of the L symbols C[c[0]], C[c[1]], . . . , C[c[L−1]].

The order in which Gaussian elimination is performed to form thedecoding schedule has no bearing on whether or not the decoding issuccessful. However, the speed of the decoding depends heavily on theorder in which Gaussian elimination is performed. (Furthermore,maintaining a sparse representation of A is crucial, although this isnot described here). The remainder of this section describes an order inwhich Gaussian elimination could be performed that is relativelyefficient.

B.6.2.2 First Phase

The first phase of the Gaussian elimination the matrix A is conceptuallypartitioned into submatrices. The submatrix sizes are parameterized bynon-negative integers i and u which are initialized to 0. Thesubmatrices of A are:

(1) The submatrix I defined by the intersection of the first i rows andfirst i columns. This is the identity matrix at the end of each step inthe phase.(2) The submatrix defined by the intersection of the first i rows andall but the first i columns and last u columns. All entries of thissubmatrix are zero.(3) The submatrix defined by the intersection of the first i columns andall but the first i rows. All entries of this submatrix are zero.(4) The submatrix U defined by the intersection of all the rows and thelast u columns.(5) The submatrix V formed by the intersection of all but the first icolumns and the last u columns and all but the first i rows.

FIG. 22 illustrates the submatrices of A. At the beginning of the firstphase V=A. In each step, a row of A is chosen. The following graphdefined by the structure of V is used in determining which row of A ischosen. The columns that intersect V are the nodes in the graph, and therows that have exactly 2 ones in V are the edges of the graph thatconnect the two columns (nodes) in the positions of the two ones. Acomponent in this graph is a maximal set of nodes (columns) and edges(rows) such that there is a path between each pair of nodes/edges in thegraph. The size of a component is the number of nodes (columns) in thecomponent.

There are at most L steps in the first phase. The phase endssuccessfully when i+u=L, i.e., when V and the all zeroes submatrix aboveV have disappeared and A comprises I, the all zeroes submatrix below I,and U. The phase ends unsuccessfully in decoding failure if at some stepbefore V disappears there is no non-zero row in V to choose in thatstep. In each step, a row of A is chosen as follows:

If all entries of V are zero then no row is chosen and decoding fails.Let r be the minimum integer such that at least one row of A has exactlyr ones in V.If r≠2 then choose a row with exactly r ones in V with minimum originaldegree among all such rows.If r=2 then choose any row with exactly 2 ones in V that is part of amaximum size component in the graph defined by X.

After the row is chosen in this step the first row of A that intersectsV is exchanged with the chosen row so that the chosen row is the firstrow that intersects V. The columns of A among those that intersect V arereordered so that one of the r ones in the chosen row appears in thefirst column of V and so that the remaining r−1 ones appear in the lastcolumns of V. Then, the chosen row is exclusive-ORed into all the otherrows of A below the chosen row that have a one in the first column of V.Finally, i is incremented by 1 and u is incremented by r−1, whichcompletes the step.

B.6.2.3 Second Phase

The submatrix U is further partitioned into the first i rows, U_(upper),and the remaining M-i rows, U_(lower). Gaussian elimination is performedin the second phase on U_(lower) to either determine that its rank isless than u (decoding failure) or to convert it into a matrix where thefirst u rows is the identity matrix (success of the second phase). Callthis u by u identity matrix I_(u). The M-L rows of A that intersectU_(lower)−I_(u) are discarded. After this phase A has L rows and Lcolumns.

B.6.2.4 Third Phase

After the second phase the only portion of A which needs to be zeroedout to finish converting A into the L by L identity matrix is U_(upper).The number of rows i of the submatrix U_(upper) is generally much largerthan the number of columns u of U_(upper). To zero out U_(upper)efficiently, the following precomputation matrix U′ is computed based onI_(u) in the third phase and then U′ is used in the fourth phase to zeroout U_(upper). The u rows of I_(u) are partitioned into ceil(u/8) groupsof 8 rows each. Then, for each group of 8 rows all non-zero combinationsof the 8 rows are computed, resulting in 2⁸−1=255 rows (this can be donewith 2⁸−8−1=247 exclusive-ors of rows per group, since the combinationsof Hamming weight one that appear in I_(u) do not need to berecomputed). Thus, the resulting precomputation matrix U′ hasceil(u/8)·255 rows and u columns. Note that U′ is not formally a part ofmatrix A, but will be used in the fourth phase to zero out U_(upper).

B.6.2.5 Fourth Phase

For each of the first i rows of A, for each group of 8 columns in theU_(upper) submatrix of this row, if the set of 8 column entries inU_(upper) are not all zero then the row of the precomputation matrix U′that matches the pattern in the 8 columns is exclusive-ORed into therow, thus zeroing out those 8 columns in the row at the cost ofexclusive-oring one row of U′ into the row.

After this phase A is the L by L identity matrix and a complete decodingschedule has been successfully formed. Then, the corresponding decodingcomprising exclusive-ORing known encoding symbols can be executed torecover the intermediate symbols based on the decoding schedule.

The triples associated with all source symbols are computed according toB.5.2.2 The triples for received source symbols are used in thedecoding. The triples for missing source symbols are used to determinewhich intermediate symbols need to be exclusive-ORed to recover themissing source symbols.

Some Properties of Some Multi-Stage Codes

In most of the examples described above, the input and output symbolsencode 98 for the same number of bits and each output symbol is placedin one packet (a packet being a unit of transport that is eitherreceived in its entirety or lost in its entirety). In some embodiments,the communications system is modified so that each packet containsseveral output symbols. The size of an output symbol value is then setto a size determined by the size of the input symbol values in theinitial splitting of the file or blocks of the stream into inputsymbols, based on a number of factors. The decoding process remainsessentially unchanged, except that output symbols arrive in bunches aseach packet is received.

The setting of input symbol and output symbol sizes is usually dictatedby the size of the file or block of the stream and the communicationsystem over which the output symbols are to be transmitted. For example,if a communication system groups bits of data into packets of a definedsize or groups bits in other ways, the design of symbol sizes beginswith the packet or grouping size. From there, a designer would determinehow many output symbols will be carried in one packet or group and thatdetermines the output symbol size. For simplicity, the designer wouldlikely set the input symbol size equal to the output symbol size, but ifthe input data makes a different input symbol size more convenient, itcan be used.

The above-described encoding process produces a stream of packetscontaining output symbols based on the original file or block of thestream. Each output symbol in the stream is generated independently ofall other output symbols, and there is no lower or upper bound on thenumber of output symbols that can be created. A key is associated witheach output symbol. That key, and some contents of the input file orblock of the stream, determines the value of the output symbol.Consecutively generated output symbols need not have consecutive keys,and in some applications it would be preferable to randomly generate thesequence of keys, or pseudorandomly generate the sequence.

Multi-stage decoding has a property that if the original file or blockof the stream can be split into K equal-sized input symbols and eachoutput symbol value is the same length as an input symbol value, thenthe file or block can be recovered from K+A output symbols on average,with very high probability, where A is small compared to K. For example,for the weight distributions introduced above, the probability that thevalue of A exceeds α*K is at most 10⁻¹² if K is larger than 19,681, andit is at most 10⁻¹⁰ for any value of K. Since the particular outputsymbols are generated in a random or pseudorandom order, and the loss ofparticular output symbols in transit is assumed random, some smallvariance exists in the actual number of output symbols needed to recoverthe input file or block. In some cases, where a particular collection ofK+A packets are not enough to decode the entire input file or block, theinput file or block is still recoverable if the receiver can gather morepackets from one or more sources of output packets.

Because the number of output symbols is only limited by the resolutionof I, well more than K+A output symbols can be generated. For example,if I is a 32-bit number, 4 billion different output symbols could begenerated, whereas the file or block of the stream could includeK=50,000 input symbols. In some applications, only a small number ofthose 4 billion output symbols may be generated and transmitted and itis a near certainty that an input file or block of a stream can berecovered with a very small fraction of the possible output symbols andan excellent probability that the input file or block can be recoveredwith slightly more than K output symbols (assuming that the input symbolsize is the same as the output symbol size).

In some applications, it may be acceptable to not be able to decode allof the input symbols, or to be able to decode all of input symbols, butwith a relatively low probability. In such applications, a receiver canstop attempting to decode all of the input symbols after receiving K+Aoutput symbols. Or, the receiver can stop receiving output symbols afterreceiving less than K+A output symbols. In some applications, thereceiver may even only receive K or less output symbols. Thus, it is tobe understood that in some embodiments of the present invention, thedesired degree of accuracy need not be complete recovery of all theinput symbols.

Further, in some applications where incomplete recovery is acceptable,the data can be encoded such that all of the input symbols cannot berecovered, or such that complete recovery of the input symbols wouldrequire reception of many more output symbols than the number of inputsymbols. Such an encoding would generally require less computationalexpense, and may thus be an acceptable way to decrease the computationalexpense of encoding.

It is to be understood that the various functional blocks in theabove-described figures may be implemented by a combination of hardwareand/or software, and that in specific implementations some or all of thefunctionality of some of the blocks may be combined. Similarly, it isalso to be understood that the various methods described herein may beimplemented by a combination of hardware and/or software.

The above description is illustrative and not restrictive. Manyvariations of the invention will become apparent to those of skill inthe art upon review of this disclosure. The scope of the inventionshould, therefore, be determined not with reference to the abovedescription, but instead should be determined with reference to theappended claims along with their full scope of equivalents.

APPENDIX A. VALUES FOR SYSTEMATIC INDICES J(K)

For each value of K, the systematic index J(K) is designed to have theproperty that the set of source symbol triples (d[0], a[0], b[0]), . . ., (d[L−1], a[L−1], b[L−1]) are such that the L intermediate symbols areuniquely defined, i.e., the matrix A in Section B.5.2.4.2 has full rankand is therefore invertible. The following is the list of the systematicindices for values of K between 4 and 8192 inclusive. The order of thevalues is in reading order, i.e., from the first number on the firstline through the last number on the first line, followed by the firstnumber on the second line, and so on.

18, 14, 61, 46, 14, 22, 20, 40, 48, 1, 29, 40, 43, 46, 18, 8, 20, 2, 61,26, 13, 29, 36, 19, 58, 5, 58, 0, 54, 56, 24, 14, 5, 67, 39, 31, 25, 29,24, 19, 14, 56, 49, 49, 63, 30, 4, 39, 2, 1, 20, 19, 61, 4, 54, 70, 25,52, 9, 26, 55, 69, 27, 68, 75, 19, 64, 57, 45, 3, 37, 31, 100, 41, 25,41, 53, 23, 9, 31, 26, 30, 30, 46, 90, 50, 13, 90, 77, 61, 31, 54, 54,3, 21, 66, 21, 11, 23, 11, 29, 21, 7, 1, 27, 4, 34, 17, 85, 69, 17, 75,93, 57, 0, 53, 71, 88, 119, 88, 90, 22, 0, 58, 41, 22, 96, 26, 79, 118,19, 3, 81, 72, 50, 0, 32, 79, 28, 25, 12, 25, 29, 3, 37, 30, 30, 41, 84,32, 31, 61, 32, 61, 7, 56, 54, 39, 33, 66, 29, 3, 14, 75, 75, 78, 84,75, 84, 25, 54, 25, 25, 107, 78, 27, 73, 0, 49, 96, 53, 50, 21, 10, 73,58, 65, 27, 3, 27, 18, 54, 45, 69, 29, 3, 65, 31, 71, 76, 56, 54, 76,54, 13, 5, 18, 142, 17, 3, 37, 114, 41, 25, 56, 0, 23, 3, 41, 22, 22,31, 18, 48, 31, 58, 37, 75, 88, 3, 56, 1, 95, 19, 73, 52, 52, 4, 75, 26,1, 25, 10, 1, 70, 31, 31, 12, 10, 54, 46, 11, 74, 84, 74, 8, 58, 23, 74,8, 36, 11, 16, 94, 76, 14, 57, 65, 8, 22, 10, 36, 36, 96, 62, 103, 6,75, 103, 58, 10, 15, 41, 75, 125, 58, 15, 10, 34, 29, 34, 4, 16, 29, 18,18, 28, 71, 28, 43, 77, 18, 41, 41, 41, 62, 29, 96, 15, 106, 43, 15, 3,43, 61, 3, 18, 103, 77, 29, 103, 19, 58, 84, 58, 1, 146, 32, 3, 70, 52,54, 29, 70, 69, 124, 62, 1, 26, 38, 26, 3, 16, 26, 5, 51, 120, 41, 16,1, 43, 34, 34, 29, 37, 56, 29, 96, 86, 54, 25, 84, 50, 34, 34, 93, 84,96, 29, 29, 50, 50, 6, 1, 105, 78, 15, 37, 19, 50, 71, 36, 6, 54, 8, 28,54, 75, 75, 16, 75, 131, 5, 25, 16, 69, 17, 69, 6, 96, 53, 96, 41, 119,6, 6, 88, 50, 88, 52, 37, 0, 124, 73, 73, 7, 14, 36, 69, 79, 6, 114, 40,79, 17, 77, 24, 44, 37, 69, 27, 37, 29, 33, 37, 50, 31, 69, 29, 101, 7,61, 45, 17, 73, 37, 34, 18, 94, 22, 22, 63, 3, 25, 25, 17, 3, 90, 34,34, 41, 34, 41, 54, 41, 54, 41, 41, 41, 163, 143, 96, 18, 32, 39, 86,104, 11, 17, 17, 11, 86, 104, 78, 70, 52, 78, 17, 73, 91, 62, 7, 128,50, 124, 18, 101, 46, 10, 75, 104, 73, 58, 132, 34, 13, 4, 95, 88, 33,76, 74, 54, 62, 113, 114, 103, 32, 103, 69, 54, 53, 3, 11, 72, 31, 53,102, 37, 53, 11, 81, 41, 10, 164, 10, 41, 31, 36, 113, 82, 3, 125, 62,16, 4, 41, 41, 4, 128, 49, 138, 128, 74, 103, 0, 6, 101, 41, 142, 171,39, 105, 121, 81, 62, 41, 81, 37, 3, 81, 69, 62, 3, 69, 70, 21, 29, 4,91, 87, 37, 79, 36, 21, 71, 37, 41, 75, 128, 128, 15, 25, 3, 108, 73,91, 62, 114, 62, 62, 36, 36, 15, 58, 114, 61, 114, 58, 105, 114, 41, 61,176, 145, 46, 37, 30, 220, 77, 138, 15, 1, 128, 53, 50, 50, 58, 8, 91,114, 105, 63, 91, 37, 37, 13, 169, 51, 102, 6, 102, 23, 105, 23, 58, 6,29, 29, 19, 82, 29, 13, 36, 27, 29, 61, 12, 18, 127, 127, 12, 44, 102,18, 4, 15, 206, 53, 127, 53, 17, 69, 69, 69, 29, 29, 109, 25, 102, 25,53, 62, 99, 62, 62, 29, 62, 62, 45, 91, 125, 29, 29, 29, 4, 117, 72, 4,30, 71, 71, 95, 79, 179, 71, 30, 53, 32, 32, 49, 25, 91, 25, 26, 26,103, 123, 26, 41, 162, 78, 52, 103, 25, 6, 142, 94, 45, 45, 94, 127, 94,94, 94, 47, 209, 138, 39, 39, 19, 154, 73, 67, 91, 27, 91, 84, 4, 84,91, 12, 14, 165, 142, 54, 69, 192, 157, 185, 8, 95, 25, 62, 103, 103,95, 71, 97, 62, 128, 0, 29, 51, 16, 94, 16, 16, 51, 0, 29, 85, 10, 105,16, 29, 29, 13, 29, 4, 4, 132, 23, 95, 25, 54, 41, 29, 50, 70, 58, 142,72, 70, 15, 72, 54, 29, 22, 145, 29, 127, 29, 85, 58, 101, 34, 165, 91,46, 46, 25, 185, 25, 77, 128, 46, 128, 46, 188, 114, 46, 25, 45, 45,114, 145, 114, 15, 102, 142, 8, 73, 31, 139, 157, 13, 79, 13, 114, 150,8, 90, 91, 123, 69, 82, 132, 8, 18, 10, 102, 103, 114, 103, 8, 103, 13,115, 55, 62, 3, 8, 154, 114, 99, 19, 8, 31, 73, 19, 99, 10, 6, 121, 32,13, 32, 119, 32, 29, 145, 30, 13, 13, 114, 145, 32, 1, 123, 39, 29, 31,69, 31, 140, 72, 72, 25, 25, 123, 25, 123, 8, 4, 85, 8, 25, 39, 25, 39,85, 138, 25, 138, 25, 33, 102, 70, 25, 25, 31, 25, 25, 192, 69, 69, 114,145, 120, 120, 8, 33, 98, 15, 212, 155, 8, 101, 8, 8, 98, 68, 155, 102,132, 120, 30, 25, 123, 123, 101, 25, 123, 32, 24, 94, 145, 32, 24, 94,118, 145, 101, 53, 53, 25, 128, 173, 142, 81, 81, 69, 33, 33, 125, 4, 1,17, 27, 4, 17, 102, 27, 13, 25, 128, 71, 13, 39, 53, 13, 53, 47, 39, 23,128, 53, 39, 47, 39, 135, 158, 136, 36, 36, 27, 157, 47, 76, 213, 47,156, 25, 25, 53, 25, 53, 25, 86, 27, 159, 25, 62, 79, 39, 79, 25, 145,49, 25, 143, 13, 114, 150, 130, 94, 102, 39, 4, 39, 61, 77, 228, 22, 25,47, 119, 205, 122, 119, 205, 119, 22, 119, 258, 143, 22, 81, 179, 22,22, 143, 25, 65, 53, 168, 36, 79, 175, 37, 79, 70, 79, 103, 70, 25, 175,4, 96, 96, 49, 128, 138, 96, 22, 62, 47, 95, 105, 95, 62, 95, 62, 142,103, 69, 103, 30, 103, 34, 173, 127, 70, 127, 132, 18, 85, 22, 71, 18,206, 206, 18, 128, 145, 70, 193, 188, 8, 125, 114, 70, 128, 114, 145,102, 25, 12, 108, 102, 94, 10, 102, 1, 102, 124, 22, 22, 118, 132, 22,116, 75, 41, 63, 41, 189, 208, 55, 85, 69, 8, 71, 53, 71, 69, 102, 165,41, 99, 69, 33, 33, 29, 156, 102, 13, 251, 102, 25, 13, 109, 102, 164,102, 164, 102, 25, 29, 228, 29, 259, 179, 222, 95, 94, 30, 30, 30, 142,55, 142, 72, 55, 102, 128, 17, 69, 164, 165, 3, 164, 36, 165, 27, 27,45, 21, 21, 237, 113, 83, 231, 106, 13, 154, 13, 154, 128, 154, 148,258, 25, 154, 128, 3, 27, 10, 145, 145, 21, 146, 25, 1, 185, 121, 0, 1,95, 55, 95, 95, 30, 0, 27, 95, 0, 95, 8, 222, 27, 121, 30, 95, 121, 0,98, 94, 131, 55, 95, 95, 30, 98, 30, 0, 91, 145, 66, 179, 66, 58, 175,29, 0, 31, 173, 146, 160, 39, 53, 28, 123, 199, 123, 175, 146, 156, 54,54, 149, 25, 70, 178, 128, 25, 70, 70, 94, 224, 54, 4, 54, 54, 25, 228,160, 206, 165, 143, 206, 108, 220, 234, 160, 13, 169, 103, 103, 103, 91,213, 222, 91, 103, 91, 103, 31, 30, 123, 13, 62, 103, 50, 106, 42, 13,145, 114, 220, 65, 8, 8, 175, 11, 104, 94, 118, 132, 27, 118, 193, 27,128, 127, 127, 183, 33, 30, 29, 103, 128, 61, 234, 165, 41, 29, 193, 33,207, 41, 165, 165, 55, 81, 157, 157, 8, 81, 11, 27, 8, 8, 98, 96, 142,145, 41, 179, 112, 62, 180, 206, 206, 165, 39, 241, 45, 151, 26, 197,102, 192, 125, 128, 67, 128, 69, 128, 197, 33, 125, 102, 13, 103, 25,30, 12, 30, 12, 30, 25, 77, 12, 25, 180, 27, 10, 69, 235, 228, 343, 118,69, 41, 8, 69, 175, 25, 69, 25, 125, 41, 25, 41, 8, 155, 146, 155, 146,155, 206, 168, 128, 157, 27, 273, 211, 211, 168, 11, 173, 154, 77, 173,77, 102, 102, 102, 8, 85, 95, 102, 157, 28, 122, 234, 122, 157, 235,222, 241, 10, 91, 179, 25, 13, 25, 41, 25, 206, 41, 6, 41, 158, 206,206, 33, 296, 296, 33, 228, 69, 8, 114, 148, 33, 29, 66, 27, 27, 30,233, 54, 173, 108, 106, 108, 108, 53, 103, 33, 33, 33, 176, 27, 27, 205,164, 105, 237, 41, 27, 72, 165, 29, 29, 259, 132, 132, 132, 364, 71, 71,27, 94, 160, 127, 51, 234, 55, 27, 95, 94, 165, 55, 55, 41, 0, 41, 128,4, 123, 173, 6, 164, 157, 121, 121, 154, 86, 164, 164, 25, 93, 164, 25,164, 210, 284, 62, 93, 30, 25, 25, 30, 30, 260, 130, 25, 125, 57, 53,166, 166, 166, 185, 166, 158, 94, 113, 215, 159, 62, 99, 21, 172, 99,184, 62, 259, 4, 21, 21, 77, 62, 173, 41, 146, 6, 41, 128, 121, 41, 11,121, 103, 159, 164, 175, 206, 91, 103, 164, 72, 25, 129, 72, 206, 129,33, 103, 102, 102, 29, 13, 11, 251, 234, 135, 31, 8, 123, 65, 91, 121,129, 65, 243, 10, 91, 8, 65, 70, 228, 220, 243, 91, 10, 10, 30, 178, 91,178, 33, 21, 25, 235, 165, 11, 161, 158, 27, 27, 30, 128, 75, 36, 30,36, 36, 173, 25, 33, 178, 112, 162, 112, 112, 112, 162, 33, 33, 178,123, 123, 39, 106, 91, 106, 106, 158, 106, 106, 284, 39, 230, 21, 228,11, 21, 228, 159, 241, 62, 10, 62, 10, 68, 234, 39, 39, 138, 62, 22, 27,183, 22, 215, 10, 175, 175, 353, 228, 42, 193, 175, 175, 27, 98, 27,193, 150, 27, 173, 17, 233, 233, 25, 102, 123, 152, 242, 108, 4, 94,176, 13, 41, 219, 17, 151, 22, 103, 103, 53, 128, 233, 284, 25, 265,128, 39, 39, 138, 42, 39, 21, 86, 95, 127, 29, 91, 46, 103, 103, 215,25, 123, 123, 230, 25, 193, 180, 30, 60, 30, 242, 136, 180, 193, 30,206, 180, 60, 165, 206, 193, 165, 123, 164, 103, 68, 25, 70, 91, 25, 82,53, 82, 186, 53, 82, 53, 25, 30, 282, 91, 13, 234, 160, 160, 126, 149,36, 36, 160, 149, 178, 160, 39, 294, 149, 149, 160, 39, 95, 221, 186,106, 178, 316, 267, 53, 53, 164, 159, 164, 165, 94, 228, 53, 52, 178,183, 53, 294, 128, 55, 140, 294, 25, 95, 366, 15, 304, 13, 183, 77, 230,6, 136, 235, 121, 311, 273, 36, 158, 235, 230, 98, 201, 165, 165, 165,91, 175, 248, 39, 185, 128, 39, 39, 128, 313, 91, 36, 219, 130, 25, 130,234, 234, 130, 234, 121, 205, 304, 94, 77, 64, 259, 60, 60, 60, 77, 242,60, 145, 95, 270, 18, 91, 199, 159, 91, 235, 58, 249, 26, 123, 114, 29,15, 191, 15, 30, 55, 55, 347, 4, 29, 15, 4, 341, 93, 7, 30, 23, 7, 121,266, 178, 261, 70, 169, 25, 25, 158, 169, 25, 169, 270, 270, 13, 128,327, 103, 55, 128, 103, 136, 159, 103, 327, 41, 32, 111, 111, 114, 173,215, 173, 25, 173, 180, 114, 173, 173, 98, 93, 25, 160, 157, 159, 160,159, 159, 160, 320, 35, 193, 221, 33, 36, 136, 248, 91, 215, 125, 215,156, 68, 125, 125, 1, 287, 123, 94, 30, 184, 13, 30, 94, 123, 206, 12,206, 289, 128, 122, 184, 128, 289, 178, 29, 26, 206, 178, 65, 206, 128,192, 102, 197, 36, 94, 94, 155, 10, 36, 121, 280, 121, 368, 192, 121,121, 179, 121, 36, 54, 192, 121, 192, 197, 118, 123, 224, 118, 10, 192,10, 91, 269, 91, 49, 206, 184, 185, 62, 8, 49, 289, 30, 5, 55, 30, 42,39, 220, 298, 42, 347, 42, 234, 42, 70, 42, 55, 321, 129, 172, 173, 172,13, 98, 129, 325, 235, 284, 362, 129, 233, 345, 175, 261, 175, 60, 261,58, 289, 99, 99, 99, 206, 99, 36, 175, 29, 25, 432, 125, 264, 168, 173,69, 158, 273, 179, 164, 69, 158, 69, 8, 95, 192, 30, 164, 101, 44, 53,273, 335, 273, 53, 45, 128, 45, 234, 123, 105, 103, 103, 224, 36, 90,211, 282, 264, 91, 228, 91, 166, 264, 228, 398, 50, 101, 91, 264, 73,36, 25, 73, 50, 50, 242, 36, 36, 58, 165, 204, 353, 165, 125, 320, 128,298, 298, 180, 128, 60, 102, 30, 30, 53, 179, 234, 325, 234, 175, 21,250, 215, 103, 21, 21, 250, 91, 211, 91, 313, 301, 323, 215, 228, 160,29, 29, 81, 53, 180, 146, 248, 66, 159, 39, 98, 323, 98, 36, 95, 218,234, 39, 82, 82, 230, 62, 13, 62, 230, 13, 30, 98, 0, 8, 98, 8, 98, 91,267, 121, 197, 30, 78, 27, 78, 102, 27, 298, 160, 103, 264, 264, 264,175, 17, 273, 273, 165, 31, 160, 17, 99, 17, 99, 234, 31, 17, 99, 36,26, 128, 29, 214, 353, 264, 102, 36, 102, 264, 264, 273, 273, 4, 16,138, 138, 264, 128, 313, 25, 420, 60, 10, 280, 264, 60, 60, 103, 178,125, 178, 29, 327, 29, 36, 30, 36, 4, 52, 183, 183, 173, 52, 31, 173,31, 158, 31, 158, 31, 9, 31, 31, 353, 31, 353, 173, 415, 9, 17, 222, 31,103, 31, 165, 27, 31, 31, 165, 27, 27, 206, 31, 31, 4, 4, 30, 4, 4, 264,185, 159, 310, 273, 310, 173, 40, 4, 173, 4, 173, 4, 250, 250, 62, 188,119, 250, 233, 62, 121, 105, 105, 54, 103, 111, 291, 236, 236, 103, 297,36, 26, 316, 69, 183, 158, 206, 129, 160, 129, 184, 55, 179, 279, 11,179, 347, 160, 184, 129, 179, 351, 179, 353, 179, 129, 129, 351, 11,111, 93, 93, 235, 103, 173, 53, 93, 50, 111, 86, 123, 94, 36, 183, 60,55, 55, 178, 219, 253, 321, 178, 235, 235, 183, 183, 204, 321, 219, 160,193, 335, 121, 70, 69, 295, 159, 297, 231, 121, 231, 136, 353, 136, 121,279, 215, 366, 215, 353, 159, 353, 353, 103, 31, 31, 298, 298, 30, 30,165, 273, 25, 219, 35, 165, 259, 54, 36, 54, 54, 165, 71, 250, 327, 13,289, 165, 196, 165, 165, 94, 233, 165, 94, 60, 165, 96, 220, 166, 271,158, 397, 122, 53, 53, 137, 280, 272, 62, 30, 30, 30, 105, 102, 67, 140,8, 67, 21, 270, 298, 69, 173, 298, 91, 179, 327, 86, 179, 88, 179, 179,55, 123, 220, 233, 94, 94, 175, 13, 53, 13, 154, 191, 74, 83, 83, 325,207, 83, 74, 83, 325, 74, 316, 388, 55, 55, 364, 55, 183, 434, 273, 273,273, 164, 213, 11, 213, 327, 321, 21, 352, 185, 103, 13, 13, 55, 30,323, 123, 178, 435, 178, 30, 175, 175, 30, 481, 527, 175, 125, 232, 306,232, 206, 306, 364, 206, 270, 206, 232, 10, 30, 130, 160, 130, 347, 240,30, 136, 130, 347, 136, 279, 298, 206, 30, 103, 273, 241, 70, 206, 306,434, 206, 94, 94, 156, 161, 321, 321, 64, 161, 13, 183, 183, 83, 161,13, 169, 13, 159, 36, 173, 159, 36, 36, 230, 235, 235, 159, 159, 335,312, 42, 342, 264, 39, 39, 39, 34, 298, 36, 36, 252, 164, 29, 493, 29,387, 387, 435, 493, 132, 273, 105, 132, 74, 73, 206, 234, 273, 206, 95,15, 280, 280, 280, 280, 397, 273, 273, 242, 397, 280, 397, 397, 397,273, 397, 280, 230, 137, 353, 67, 81, 137, 137, 353, 259, 312, 114, 164,164, 25, 77, 21, 77, 165, 30, 30, 231, 234, 121, 234, 312, 121, 364,136, 123, 123, 136, 123, 136, 150, 264, 285, 30, 166, 93, 30, 39, 224,136, 39, 355, 355, 397, 67, 67, 25, 67, 25, 298, 11, 67, 264, 374, 99,150, 321, 67, 70, 67, 295, 150, 29, 321, 150, 70, 29, 142, 355, 311,173, 13, 253, 103, 114, 114, 70, 192, 22, 128, 128, 183, 184, 70, 77,215, 102, 292, 30, 123, 279, 292, 142, 33, 215, 102, 468, 123, 468, 473,30, 292, 215, 30, 213, 443, 473, 215, 234, 279, 279, 279, 279, 265, 443,206, 66, 313, 34, 30, 206, 30, 51, 15, 206, 41, 434, 41, 398, 67, 30,301, 67, 36, 3, 285, 437, 136, 136, 22, 136, 145, 365, 323, 323, 145,136, 22, 453, 99, 323, 353, 9, 258, 323, 231, 128, 231, 382, 150, 420,39, 94, 29, 29, 353, 22, 22, 347, 353, 39, 29, 22, 183, 8, 284, 355,388, 284, 60, 64, 99, 60, 64, 150, 95, 150, 364, 150, 95, 150, 6, 236,383, 544, 81, 206, 388, 206, 58, 159, 99, 231, 228, 363, 363, 121, 99,121, 121, 99, 422, 544, 273, 173, 121, 427, 102, 121, 235, 284, 179, 25,197, 25, 179, 511, 70, 368, 70, 25, 388, 123, 368, 159, 213, 410, 159,236, 127, 159, 21, 373, 184, 424, 327, 250, 176, 176, 175, 284, 316,176, 284, 327, 111, 250, 284, 175, 175, 264, 111, 176, 219, 111, 427,427, 176, 284, 427, 353, 428, 55, 184, 493, 158, 136, 99, 287, 264, 334,264, 213, 213, 292, 481, 93, 264, 292, 295, 295, 6, 367, 279, 173, 308,285, 158, 308, 335, 299, 137, 137, 572, 41, 137, 137, 41, 94, 335, 220,36, 224, 420, 36, 265, 265, 91, 91, 71, 123, 264, 91, 91, 123, 107, 30,22, 292, 35, 241, 356, 298, 14, 298, 441, 35, 121, 71, 63, 130, 63, 488,363, 71, 63, 307, 194, 71, 71, 220, 121, 125, 71, 220, 71, 71, 71, 71,235, 265, 353, 128, 155, 128, 420, 400, 130, 173, 183, 183, 184, 130,173, 183, 13, 183, 130, 130, 183, 183, 353, 353, 183, 242, 183, 183,306, 324, 324, 321, 306, 321, 6, 6, 128, 306, 242, 242, 306, 183, 183,6, 183, 321, 486, 183, 164, 30, 78, 138, 158, 138, 34, 206, 362, 55, 70,67, 21, 375, 136, 298, 81, 298, 298, 298, 230, 121, 30, 230, 311, 240,311, 311, 158, 204, 136, 136, 184, 136, 264, 311, 311, 312, 312, 72,311, 175, 264, 91, 175, 264, 121, 461, 312, 312, 238, 475, 350, 512,350, 312, 313, 350, 312, 366, 294, 30, 253, 253, 253, 388, 158, 388, 22,388, 22, 388, 103, 321, 321, 253, 7, 437, 103, 114, 242, 114, 114, 242,114, 114, 242, 242, 242, 306, 242, 114, 7, 353, 335, 27, 241, 299, 312,364, 506, 409, 94, 462, 230, 462, 243, 230, 175, 175, 462, 461, 230,428, 426, 175, 175, 165, 175, 175, 372, 183, 572, 102, 85, 102, 538,206, 376, 85, 85, 284, 85, 85, 284, 398, 83, 160, 265, 308, 398, 310,583, 289, 279, 273, 285, 490, 490, 211, 292, 292, 158, 398, 30, 220,169, 368, 368, 368, 169, 159, 368, 93, 368, 368, 93, 169, 368, 368, 443,368, 298, 443, 368, 298, 538, 345, 345, 311, 178, 54, 311, 215, 178,175, 222, 264, 475, 264, 264, 475, 478, 289, 63, 236, 63, 299, 231, 296,397, 299, 158, 36, 164, 164, 21, 492, 21, 164, 21, 164, 403, 26, 26,588, 179, 234, 169, 465, 295, 67, 41, 353, 295, 538, 161, 185, 306, 323,68, 420, 323, 82, 241, 241, 36, 53, 493, 301, 292, 241, 250, 63, 63,103, 442, 353, 185, 353, 321, 353, 185, 353, 353, 185, 409, 353, 589,34, 271, 271, 34, 86, 34, 34, 353, 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398,1027, 398, 295, 784, 740, 509, 425, 968, 509, 46, 833, 842, 401, 184,401, 464, 6, 1501, 1501, 550, 538, 883, 538, 883, 883, 883, 1129, 550,550, 333, 689, 948, 21, 21, 241, 2557, 2094, 273, 308, 58, 863, 893,1086, 409, 136, 1086, 592, 592, 830, 830, 883, 830, 277, 68, 689, 902,277, 453, 507, 129, 689, 630, 664, 550, 128, 1626, 1626, 128, 902, 312,589, 755, 755, 589, 755, 407, 1782, 589, 784, 1516, 1118, 407, 407,1447, 589, 235, 755, 1191, 235, 235, 407, 128, 589, 1118, 21, 383, 1331,691, 481, 383, 1129, 1129, 1261, 1104, 1378, 1129, 784, 1129, 1261,1129, 947, 1129, 784, 784, 1129, 1129, 35, 1104, 35, 866, 1129, 1129,64, 481, 730, 1260, 481, 970, 481, 481, 481, 481, 863, 481, 681, 699,863, 486, 681, 481, 481, 55, 55, 235, 1364, 944, 632, 822, 401, 822,952, 822, 822, 99, 550, 2240, 550, 70, 891, 860, 860, 550, 550, 916,1176, 1530, 425, 1530, 916, 628, 1583, 916, 628, 916, 916, 628, 628,425, 916, 1062, 1265, 916, 916, 916, 280, 461, 916, 916, 1583, 628,1062, 916, 916, 677, 1297, 924, 1260, 83, 1260, 482, 433, 234, 462, 323,1656, 997, 323, 323, 931, 838, 931, 1933, 1391, 367, 323, 931, 1391,1391, 103, 1116, 1116, 1116, 769, 1195, 1218, 312, 791, 312, 741, 791,997, 312, 334, 334, 312, 287, 287, 633, 1397, 1426, 605, 1431, 327, 592,705, 1194, 592, 1097, 1118, 1503, 1267, 1267, 1267, 618, 1229, 734,1089, 785, 1089, 1129, 1148, 1148, 1089, 915, 1148, 1129, 1148, 1011,1011, 1229, 871, 1560, 1560, 1560, 563, 1537, 1009, 1560, 632, 985, 592,1308, 592, 882, 145, 145, 397, 837, 383, 592, 592, 832, 36, 2714, 2107,1588, 1347, 36, 36, 1443, 1453, 334, 2230, 1588, 1169, 650, 1169, 2107,425, 425, 891, 891, 425, 2532, 679, 274, 274, 274, 325, 274, 1297, 194,1297, 627, 314, 917, 314, 314, 1501, 414, 1490, 1036, 592, 1036, 1025,901, 1218, 1025, 901, 280, 592, 592, 901, 1461, 159, 159, 159, 2076,1066, 1176, 1176, 516, 327, 516, 1179, 1176, 899, 1176, 1176, 323, 1187,1229, 663, 1229, 504, 1229, 916, 1229, 916, 1661, 41, 36, 278, 1027,648, 648, 648, 1626, 648, 646, 1179, 1580, 1061, 1514, 1008, 1741, 2076,1514, 1008, 952, 1089, 427, 952, 427, 1083, 425, 427, 1089, 1083, 425,427, 425, 230, 920, 1678, 920, 1678, 189, 189, 953, 189, 133, 189, 1075,189, 189, 133, 1264, 725, 189, 1629, 189, 808, 230, 230, 2179, 770, 230,770, 230, 21, 21, 784, 1118, 230, 230, 230, 770, 1118, 986, 808, 916,30, 327, 918, 679, 414, 916, 1165, 1355, 916, 755, 733, 433, 1490, 433,433, 433, 605, 433, 433, 433, 1446, 679, 206, 433, 21, 2452, 206, 206,433, 1894, 206, 822, 206, 2073, 206, 206, 21, 822, 21, 206, 206, 21,383, 1513, 375, 1347, 432, 1589, 172, 954, 242, 1256, 1256, 1248, 1256,1256, 1248, 1248, 1256, 842, 13, 592, 13, 842, 1291, 592, 21, 175, 13,592, 13, 13, 1426, 13, 1541, 445, 808, 808, 863, 647, 219, 1592, 1029,1225, 917, 1963, 1129, 555, 1313, 550, 660, 550, 220, 660, 552, 663,220, 533, 220, 383, 550, 1278, 1495, 636, 842, 1036, 425, 842, 425,1537, 1278, 842, 554, 1508, 636, 554, 301, 842, 792, 1392, 1021, 284,1172, 997, 1021, 103, 1316, 308, 1210, 848, 848, 1089, 1089, 848, 848,67, 1029, 827, 1029, 2078, 827, 1312, 1029, 827, 590, 872, 1312, 427,67, 67, 67, 67, 872, 827, 872, 2126, 1436, 26, 2126, 67, 1072, 2126,1610, 872, 1620, 883, 883, 1397, 1189, 555, 555, 563, 1189, 555, 640,555, 640, 1089, 1089, 610, 610, 1585, 610, 1355, 610, 1015, 616, 925,1015, 482, 230, 707, 231, 888, 1355, 589, 1379, 151, 931, 1486, 1486,393, 235, 960, 590, 235, 960, 422, 142, 285, 285, 327, 327, 442, 2009,822, 445, 822, 567, 888, 2611, 1537, 323, 55, 1537, 323, 888, 2611, 323,1537, 323, 58, 445, 593, 2045, 593, 58, 47, 770, 842, 47, 47, 842, 842,648, 2557, 173, 689, 2291, 1446, 2085, 2557, 2557, 2291, 1780, 1535,2291, 2391, 808, 691, 1295, 1165, 983, 948, 2000, 948, 983, 983, 2225,2000, 983, 983, 705, 948, 2000, 1795, 1592, 478, 592, 1795, 1795, 663,478, 1790, 478, 592, 1592, 173, 901, 312, 4, 1606, 173, 838, 754, 754,128, 550, 1166, 551, 1480, 550, 550, 1875, 1957, 1166, 902, 1875, 550,550, 551, 2632, 551, 1875, 1875, 551, 2891, 2159, 2632, 3231, 551, 815,150, 1654, 1059, 1059, 734, 770, 555, 1592, 555, 2059, 770, 770, 1803,627, 627, 627, 2059, 931, 1272, 427, 1606, 1272, 1606, 1187, 1204, 397,822, 21, 1645, 263, 263, 822, 263, 1645, 280, 263, 605, 1645, 2014, 21,21, 1029, 263, 1916, 2291, 397, 397, 496, 270, 270, 1319, 264, 1638,264, 986, 1278, 1397, 1278, 1191, 409, 1191, 740, 1191, 754, 754, 387,63, 948, 666, 666, 1198, 548, 63, 1248, 285, 1248, 169, 1248, 1248, 285,918, 224, 285, 1426, 1671, 514, 514, 717, 514, 51, 1521, 1745, 51, 605,1191, 51, 128, 1191, 51, 51, 1521, 267, 513, 952, 966, 1671, 897, 51,71, 592, 986, 986, 1121, 592, 280, 2000, 2000, 1165, 1165, 1165, 1818,222, 1818, 1165, 1252, 506, 327, 443, 432, 1291, 1291, 2755, 1413, 520,1318, 227, 1047, 828, 520, 347, 1364, 136, 136, 452, 457, 457, 132, 457,488, 1087, 1013, 2225, 32, 1571, 2009, 483, 67, 483, 740, 740, 1013,2854, 866, 32, 2861, 866, 887, 32, 2444, 740, 32, 32, 866, 2225, 866,32, 1571, 2627, 32, 850, 1675, 569, 1158, 32, 1158, 1797, 2641, 1565,1158, 569, 1797, 1158, 1797, 55, 1703, 42, 55, 2562, 675, 1703, 42, 55,749, 488, 488, 347, 1206, 1286, 1286, 488, 488, 1206, 1286, 1206, 1286,550, 550, 1790, 860, 550, 2452, 550, 550, 2765, 1089, 1633, 797, 2244,1313, 194, 2129, 194, 194, 194, 818, 32, 194, 450, 1313, 2387, 194,1227, 2387, 308, 2232, 526, 476, 278, 830, 830, 194, 830, 194, 278, 194,714, 476, 830, 714, 830, 278, 830, 2532, 1218, 1759, 1446, 960, 1747,187, 1446, 1759, 960, 105, 1446, 1446, 1271, 1446, 960, 960, 1218, 1446,1446, 105, 1446, 960, 488, 1446, 427, 534, 842, 1969, 2460, 1969, 842,842, 1969, 427, 941, 2160, 427, 230, 938, 2075, 1675, 1675, 895, 1675,34, 129, 1811, 239, 749, 1957, 2271, 749, 1908, 129, 239, 239, 129, 129,2271, 2426, 1355, 1756, 194, 1583, 194, 194, 1583, 194, 1355, 194, 1628,2221, 1269, 2425, 1756, 1355, 1355, 1583, 1033, 427, 582, 30, 582, 582,935, 1444, 1962, 915, 733, 915, 938, 1962, 767, 353, 1630, 1962, 1962,563, 733, 563, 733, 353, 822, 1630, 740, 2076, 2076, 2076, 589, 589,2636, 866, 589, 947, 1528, 125, 273, 1058, 1058, 1161, 1635, 1355, 1161,1161, 1355, 1355, 650, 1206, 1206, 784, 784, 784, 784, 784, 412, 461,412, 2240, 412, 679, 891, 461, 679, 679, 189, 189, 1933, 1651, 2515,189, 1386, 538, 1386, 1386, 1187, 1386, 2423, 2601, 2285, 175, 175,2331, 194, 3079, 384, 538, 2365, 2294, 538, 2166, 1841, 3326, 1256,3923, 976, 85, 550, 550, 1295, 863, 863, 550, 1249, 550, 1759, 146,1069, 920, 2633, 885, 885, 1514, 1489, 166, 1514, 2041, 885, 2456, 885,2041, 1081, 1948, 362, 550, 94, 324, 2308, 94, 2386, 94, 550, 874, 1329,1759, 2280, 1487, 493, 493, 2099, 2599, 1431, 1086, 1514, 1086, 2099,1858, 368, 1330, 2599, 1858, 2846, 2846, 2907, 2846, 713, 713, 1854,1123, 713, 713, 3010, 1123, 3010, 538, 713, 1123, 447, 822, 555, 2011,493, 508, 2292, 555, 1736, 2135, 2704, 555, 2814, 555, 2000, 555, 555,822, 914, 327, 679, 327, 648, 537, 2263, 931, 1496, 537, 1296, 1745,1592, 1658, 1795, 650, 1592, 1745, 1745, 1658, 1592, 1745, 1592, 1745,1658, 1338, 2124, 1592, 1745, 1745, 1745, 837, 1726, 2897, 1118, 1118,230, 1118, 1118, 1118, 1388, 1748, 514, 128, 1165, 931, 514, 2974, 2041,2387, 2041, 979, 185, 36, 1269, 550, 173, 812, 36, 1165, 2676, 2562,1473, 2885, 1982, 1578, 1578, 383, 383, 2360, 383, 1578, 2360, 1584,1982, 1578, 1578, 1578, 2019, 1036, 355, 724, 2023, 205, 303, 355, 1036,1966, 355, 1036, 401, 401, 401, 830, 401, 849, 578, 401, 849, 849, 578,1776, 1123, 552, 2632, 808, 1446, 1120, 373, 1529, 1483, 1057, 893,1284, 1430, 1529, 1529, 2632, 1352, 2063, 1606, 1352, 1606, 2291, 3079,2291, 1529, 506, 838, 1606, 1606, 1352, 1529, 1529, 1483, 1529, 1606,1529, 259, 902, 259, 902, 612, 612, 284, 398, 2991, 1534, 1118, 1118,1118, 1118, 1118, 734, 284, 2224, 398, 734, 284, 734, 398, 3031, 398,734, 1707, 2643, 1344, 1477, 475, 1818, 194, 1894, 691, 1528, 1184,1207, 1501, 6, 2069, 871, 2069, 3548, 1443, 2069, 2685, 3265, 1350,3265, 2069, 2069, 128, 1313, 128, 663, 414, 1313, 414, 2000, 128, 2000,663, 1313, 699, 1797, 550, 327, 550, 1526, 699, 327, 1797, 1526, 550,550, 327, 550, 1426, 1426, 1426, 2285, 1123, 890, 728, 1707, 728, 728,327, 253, 1187, 1281, 1364, 1571, 2170, 755, 3232, 925, 1496, 2170,2170, 1125, 443, 902, 902, 925, 755, 2078, 2457, 902, 2059, 2170, 1643,1129, 902, 902, 1643, 1129, 606, 36, 103, 338, 338, 1089, 338, 338, 338,1089, 338, 36, 340, 1206, 1176, 2041, 833, 1854, 1916, 1916, 1501, 2132,1736, 3065, 367, 1934, 833, 833, 833, 2041, 3017, 2147, 818, 1397, 828,2147, 398, 828, 818, 1158, 818, 689, 327, 36, 1745, 2132, 582, 1475,189, 582, 2132, 1191, 582, 2132, 1176, 1176, 516, 2610, 2230, 2230, 64,1501, 537, 1501, 173, 2230, 2988, 1501, 2694, 2694, 537, 537, 173, 173,1501, 537, 64, 173, 173, 64, 2230, 537, 2230, 537, 2230, 2230, 2069,3142, 1645, 689, 1165, 1165, 1963, 514, 488, 1963, 1145, 235, 1145,1078, 1145, 231, 2405, 552, 21, 57, 57, 57, 1297, 1455, 1988, 2310,1885, 2854, 2014, 734, 1705, 734, 2854, 734, 677, 1988, 1660, 734, 677,734, 677, 677, 734, 2854, 1355, 677, 1397, 2947, 2386, 1698, 128, 1698,3028, 2386, 2437, 2947, 2386, 2643, 2386, 2804, 1188, 335, 746, 1187,1187, 861, 2519, 1917, 2842, 1917, 675, 1308, 234, 1917, 314, 314, 2339,2339, 2592, 2576, 902, 916, 2339, 916, 2339, 916, 2339, 916, 1089, 1089,2644, 1221, 1221, 2446, 308, 308, 2225, 2225, 3192, 2225, 555, 1592,1592, 555, 893, 555, 550, 770, 3622, 2291, 2291, 3419, 465, 250, 2842,2291, 2291, 2291, 935, 160, 1271, 308, 325, 935, 1799, 1799, 1891, 2227,1799, 1598, 112, 1415, 1840, 2014, 1822, 2014, 677, 1822, 1415, 1415,1822, 2014, 2386, 2159, 1822, 1415, 1822, 179, 1976, 1033, 179, 1840,2014, 1415, 1970, 1970, 1501, 563, 563, 563, 462, 563, 1970, 1158, 563,563, 1541, 1238, 383, 235, 1158, 383, 1278, 383, 1898, 2938, 21, 2938,1313, 2201, 2059, 423, 2059, 1313, 872, 1313, 2044, 89, 173, 3327, 1660,2044, 1623, 173, 1114, 1114, 1592, 1868, 1651, 1811, 383, 3469, 1811,1651, 869, 383, 383, 1651, 1651, 3223, 2166, 3469, 767, 383, 1811, 767,2323, 3355, 1457, 3341, 2640, 2976, 2323, 3341, 2323, 2640, 103, 103,1161, 1080, 2429, 370, 2018, 2854, 2429, 2166, 2429, 2094, 2207, 871,1963, 1963, 2023, 2023, 2336, 663, 2893, 1580, 691, 663, 705, 2046,2599, 409, 2295, 1118, 2494, 1118, 1950, 549, 2494, 2453, 2046, 2494,2453, 2046, 2453, 2046, 409, 1118, 4952, 2291, 2225, 1894, 1423, 2498,567, 4129, 1475, 1501, 795, 463, 2084, 828, 828, 232, 828, 232, 232,1818, 1818, 666, 463, 232, 220, 220, 2162, 2162, 833, 4336, 913, 35,913, 21, 2927, 886, 3037, 383, 886, 876, 1747, 383, 916, 916, 916, 2927,916, 1747, 837, 1894, 717, 423, 481, 1894, 1059, 2262, 3206, 4700, 1059,3304, 2262, 871, 1831, 871, 3304, 1059, 1158, 1934, 1158, 756, 1511, 41,978, 1934, 2603, 720, 41, 756, 41, 325, 2611, 1158, 173, 1123, 1934,1934, 1511, 2045, 2045, 2045, 1423, 3206, 3691, 2512, 3206, 2512, 2000,1811, 2504, 2504, 2611, 2437, 2437, 2437, 1455, 893, 150, 2665, 1966,605, 398, 2331, 1177, 516, 1962, 4241, 94, 1252, 760, 1292, 1962, 1373,2000, 1990, 3684, 42, 1868, 3779, 1811, 1811, 2041, 3010, 5436, 1780,2041, 1868, 1811, 1780, 1811, 1868, 1811, 2041, 1868, 1811, 5627, 4274,1811, 1868, 4602, 1811, 1811, 1474, 2665, 235, 1474, 2665

APPENDIX B.1 VALUES FOR TABLE V₀

These values represent an example set of values for Table V₀ describedin the application above, in Section B.5.4.1. Each entry is a 32-bitinteger in decimal representation. The order of the values is from thetop of the first column to the bottom of the first column, followed bythe top of the second column, and so on.

251291136 3676575285 1255426612 138853680 2213589897 39522316312581671944 4213663089 4160398285 2734638932 3370958628 33122204801521339547 3322241130 1129721478 4070167936 681232419 30418434892068983570 3187422815 123631495 307306866 420130494 22474910782859178611 3351110283 4112503940 10677091 3669524410 32843084113218676425 1158111502 515623176 1575146607 3819792700 2011642291709227802 3457502702 828029864 3557526733 774603218 27241404332115821274 3732001371 451874476 2402805061 4201101115 27201247663422026452 1740576081 1004366930 4215970289 3242576090 33709541773592838701 1843948209 4048876515 854310108 4006626915 1709429513428891132 3031661061 425973987 543812220 3702918379 37463319841909085522 325832382 1243116171 3533351328 1591258008 5109850331796851292 3928372514 1641660745 3067016507 1361682810 24627444112791443445 179350258 1433388735 129243379 1976681690 40813252722380520112 504005498 3142379587 1408671665 2280435605 39361639042032657933 2569842483 1228817808 885616073 3685256204 34193197843033268270 3917210003 616452097 3156252216 2805686246 1658118006263976645 3188863436 1854258901 3102436986 932109358 25937364732780382310 2861641019 3808671154 1982290045 2471651269 23400148313176611298 2501582075 2983082771 4291353919 1208439576 8347875543978944421 3007670818 650792940 258356309 331353807 246043949 34481047681191583883 3837963200 517858103 4016898363 683749698 30465613352075009450 3010168884 649743608 778296777 2466530435 32141812124012642001 1974987508 1399125101 2545983082 3303882142 22171880752651273766 1939403708 969168436 880813252 3756943137 23579568011692176003 2019348792 1355575717 3077882590 689605112 38682992002268075521 207231484 2054995199 715807172 1422476658 11693450682420803184 3081443129 2722736134 593093658 3250240009 3589233683895398812 191939188 1878973865 3963499681 1617557768 11410975433535520147 2526292949 2560755113 3272161958 2376261053 32770195691591602827 911182396 1771154147 2626898255 1470435941 3986158854760842409 2842106362 2554703076 3763101702 3964389521 35693086931751209208 401233789 3232409631 2695031039 2687243553 14210307901460049922 122701163 1942050155 381854665 658316681 678083952 3920852693424618399 2613828404 194065839 1064990737 782246947 13472042912761078866 3241510581 940909784 372121310 2669179716 1456668111 386252601673396780 2995604341 2434425874 883760091 301875395 5288817832045698575 2540801947 3294951678 4176141739 1712547446 23329621021384069776 1604598575 297312930 3629685652 4005368743 41235804431985308198 2137802113 1358307511 218596347 1523670218 10145705431502984205 3415381967 2708475297 2724959607 3669376622 42076128061046771089 3062518035 3728477036 861117671 2229796016 3115293053234652930

APPENDIX B.2 VALUES FOR TABLE V₁

These values represent an example set of values for Table V₁ describedin the application above, in Section B.5.4.1. Each entry is a 32-bitinteger in decimal representation. The order of the values is from thetop of the first column to the bottom of the first column, followed bythe top of the second column, and so on.

807385413 3843885867 3545667983 644189126 922673938 20430732234201106668 332038910 226475395 3877430102 3336749796 415906198 976628269307789415 3422391938 1302105833 19296841 3123492423 11961056311414347295 2278607931 2402488407 3041418372 3191691839 1971054608541015020 2137119134 2258059298 782852669 3061798054 16845642701744097284 2139377204 1608507813 830555096 372709334 5799656373243642973 1847685900 2822905141 3508252125 2037662632 32262479174069766876 167033190 1768346005 852173610 3674004636 39315486411079139428 1270451292 2681403713 2698992189 2526471011 42101267232603029534 1047144830 3453843574 766865139 3593797804 20493872732982173936 1963216666 2115084288 429192890 3891424859 9102850383509855005 4259411376 372093950 2152948345 4187576520 23584818583323683436 1779187770 4114760273 2589870048 747331248 5685121773312189287 915180310 989448887 1957348676 3736601419 2043493483754787998 3292758024 1097574450 1800276898 452421568 700503826506322719 2435697214 4012458395 2800540462 2131559305 1760107383870972145 1823982 3733109044 1308908630 1865471968 1888833893 279801981235082423 224437350 2619324712 2914085525 2023839966 17653195564065424007 564829442 4161315584 869505096 3174729780 36386659441996870325 1273113343 431161506 3762994475 1679385496 3396975933269644828 1024804023 3171962488 3431345226 4071072948 36812938161853869307 442160826 1779595665 3618966336 412536684 3393537983198349622 3068494238 2111320126 1156034077 1500703614 459426371424062773 1093955153 3823026442 3019471560 1324086311 1033448464957978696 1066971017 1351086955 2901868599 4050396853 8920105603598330293 3096933631 678860040 3302235057 1854601078 19792739373034634988 3812229107 420600373 1873407527 2079029895 254459800619936821 2868446243 2498544695 1195045909 1230942551 1119590141311689386 2694156259 1071986421 3362230798 3640121682 2590479591927339682 2712821515 159984793 3545931032 4057180909 16505557293377754595 491590373 2102949142 1575367248 183933047 21841510953993872886 2828208598 4151214153 3061444337 750918864 36818556223603378023 110249784 2067387204 2585729879 903593547 41350488963006865921 228962564 4249895712 3535062472 4293710613 39041094141832579367 1799803217 3501256572 1595995433 1192240192 772984149998007483 1780701372 946734366 895863112 499288295 2463145963 312306881899036275 1205710710 307281463 3174399083 4187322100 29971994893237929991 3549375728 101856048 640417429 3852995239 1642430184234650315 3044194711 2398693510 1904857554 3183125617 4866907513754138664 861877404 3190039692 2686640734 522074127 3277825584525584357 2394526209 146352474 4267074718 1286834489 25216600774104915256 3122860549 455810374 49993987 3029415884 666423581 1869181575

1. A method of encoding data for transmission from a source to adestination over a communications channel, wherein the data isrepresented at least by a plurality, K, of source symbols, K being thenumber of source symbols and the source symbols being in an ordered set,the method comprising: generating one or more repair symbols from theplurality of source symbols; transmitting one or more symbols over thecommunications channel, wherein the communication channel is a channelthat can introduce errors in transmissions, wherein the one or moresymbols are selected from the group consisting of source symbols andrepair symbols; determining a desired degree of accuracy for theregeneration of source symbols; determining, for each of a plurality ofsource symbols, an associated symbol relation that is a function of asystematic index, J(K), where J(K) is a function of K; determining Lintermediate symbol values according to a function of values of the Ksource symbols and the K symbol associated symbol relations associatedwith the K source symbols and a set of L-K pre-coding relations, whereinL is at least K; determining a value for each repair symbol using theassociated symbol relation for that repair symbol and the plurality of Lintermediate symbol values; wherein the encoding is such that theplurality of source symbols can be regenerated to the desired degree ofaccuracy from any predetermined number, N, of a combination of thetransmitted source symbols and transmitted repair symbols.
 2. The methodof claim 1, further comprising precalculating and storing the systematicindex J(K) in a table for each value of K.
 3. The method of claim 1,wherein the systematic index J(K) for values of K between 4 and 8192 isas provided in Appendix A.
 4. The method of claim 1, wherein the symbolrelations impose linear constraints on sets of symbols.
 5. The method ofclaim 4, wherein the linear constraints are exclusive- or constraints.6. The method of claim 1, wherein each symbol has an associated ESI thatidentifies the symbol, wherein the systematic index J(K) and a value X,wherein X is a valid ESI, determines the symbol relation for the symbolidentified by ESI X.
 7. The method of claim 6, wherein the systematicindex J(K) and the ESI X can be used to generate a triple(a[X],b[X],d[X]) that defines the symbol relation for the symbolidentified by ESI X.
 8. The method of claim 6, wherein each sourcesymbol can be identified by an integer ESI in the range 0 to K−1,inclusive.
 9. The method of claim 6, wherein each repair symbol can beidentified by an integer ESI that is at least K.
 10. The method of claim7, wherein the triple (a[X],b[X],d[X]) that defines the source symbolrelation associated with the source symbol with ESI X places a linearconstraint on the value C′[X] for source symbol with ESI X and thevalues C[0], C[1], C[L−1] of the L intermediate symbols, the linearconstraint defined by the method comprising: defining an array ofpositive integer values B[0], ..., B[J−1], wherein J   is the minimum ofd[X] and L, wherein L’ is the smallest integer   that is prime that isat least the value of L, and wherein the values   of the array ofpositive integer values B[0],...,B[J−1] are set using the   methodcomprising: setting the initial value of B[0] to b[X];   while B[0] isat least L, recalculating B[0] as the previous   value of B[0] plus a[X]modulo L’;   for values of j starting at 1 and going up to J−1 inincrements of one,     initializing the value of B[j] to B[j−1] plusa[X];     while B[j] is at least L, recalculating B[j] as the     is atleast the value previous value of B[j] plus   a[X] modulo L’.


11. The method of claim 7, further comprising: calculating a value A as(53591 + J(K) * 997) % 65521, wherein *   is the multiplication operatorand % is the modulo operator; calculating a value B as 10267 *(J(K) + 1) % 65521; calculating a value Y as (B + X * A) % 65521; anddetermining a value v, wherein v is in the ranges from 0 and 1048575,  inclusive, where v is calculated as the output of a random generator  applied to the inputs Y, 0, and
 1048575.


12. The method of claim 11 further comprising: determining d[X] based ona degree function applied to the input v.
 13. The method of claim 11further comprising: determining a[X], wherein a[X] is in the rangebetween 0 and L′−1, inclusive, wherein L′ is the smallest prime numbergreater than or equal to L, wherein L′ is calculated as one plus theoutput of the random generator applied to the inputs Y, 1 and L′−1. 14.The method of claim 11 further comprising: determining b[X], whereinb[X] is in the range between 0 and L′, inclusive, wherein b[X] iscalculated as the output of the random generator applied to the input Y,2 and L′.
 15. The method of claim 11, wherein the value of d[X] isdetermined based on v as follows: if the value of v is between 0 and10240, inclusive, then the value of d[X] is 1, if the value of v isbetween 10241 and 4911581, inclusive, then the value of d[X] is 2, ifthe value of v is between 4911582 and 712793, inclusive, then the valueof d[X] is 3, if the value of v is between 712794 and 831694, inclusive,then the value of d[X] is 4, if the value of v is between 831695 and948445, inclusive, then the value of d[X] is 10, if the value of v isbetween 948446 and 1032188, inclusive, then the value of d[X] is 11, ifthe value of v is between 1032189 and 1048575, inclusive, then the valueof d[X] is
 40.


16. The method of claim 11, wherein the random generator on inputs Y, i,and m is calculated as (V0[(Y+i) % 256]̂(V1[(floor(Y/256)+i) % 256]) % m,where % is the modulo operator, ̂ is the exclusive- or operator,/is thedivision operator, floor calculates the largest integer that is at mostthe input value, and V0 and V1 are each tables of 256 integers chosenrandomly or pseudo-randomly.
 17. The method of claim 16, wherein the 256values for table V0 is as described in Appendix B.1.
 18. The method ofclaim 16, wherein the 256 values for table V1 is as described inAppendix B.2.
 19. The method of claim 1, wherein the number L-K ofpre-coding relations comprises a first set of S pre-coding relations anda second set of H pre-coding relations, and wherein the L intermediatesymbols comprises a first set of K intermediate symbols, a second set ofS intermediate symbols, and a third set of H intermediate symbols. 20.The method of claim 19, wherein each pre-coding relation in the firstset of pre-coding relations is uniquely associated with an intermediatesymbol in the second set of intermediate symbols and is associated witha specified set of intermediate symbols among the first set ofintermediate symbols, wherein each pre-coding relation is a linearconstraint on the value of the associated intermediate symbol and thevalues of the intermediate symbols in the specified set.
 21. The methodof claim 20, wherein a linear constraint is an exclusive- or constraint.22. The method of claim 20, wherein for each first pre-coding relationamong the first set of pre-coding relations and for each secondpre-coding relation among the first set of pre-coding relations, thenumber of intermediate symbols that are both in the specified setassociated with the first pre-coding relation and in the specified setassociated with the second pre-coding relation is at most
 3. 23. Themethod of claim 20, wherein each of the intermediate symbols in thefirst set of intermediate symbols is in the specified set of exactlythree of the pre-coding relations in the first set of pre-codingrelations.
 24. The method of claim 19, wherein S is the smallest primeinteger that is at least as large as 0.01*K+X, wherein X is the smallestpositive integer such that X*(X−1) is at least 2*K, wherein H is thesmallest integer such that fact[H]/(fact[H−H′]*fact[H′]) is at leastK+2, wherein fact is the factorial operator, wherein H′ is the smallestinteger that is at least H/2.
 25. The method of claim 19, wherein eachpre-coding relation in the second set of pre-coding relations isuniquely associated with an intermediate symbol in the third set ofintermediate symbols and is associated with a specified set ofintermediate symbols among the first set and second set of intermediatesymbols, wherein the pre-coding relation is a linear constraint on theassociated intermediate symbol and the intermediate symbols in thespecified set.
 26. The method of claim 25, wherein a linear constraintis an exclusive- or constraint.
 27. The method of claim 25, wherein eachof the intermediate symbols in the first set or the second set ofintermediate symbols is in the specified set of approximately one-halfof the pre-coding relations in the second set of pre-coding relations.28. The method of claim 25, wherein for each first intermediate symbolin the first set or second set of intermediate symbols and for eachsecond intermediate symbol in the first set or second set ofintermediate symbols that is the next consecutive intermediate symbolfollowing the first intermediate symbol, the symmetric difference ofpre-coding relations in the second set of pre-coding relations of whichthe first intermediate symbol is a member of the specified set and ofwhich the second intermediate symbol is a member of the specified set isexactly two.
 29. A method of decoding encoded data received over acommunications channel transmitted from a source to a destination, themethod comprising: receiving a predetermined number, N, of symbols,wherein the symbols comprise a combination of received source symbolsand repair symbols generated from a plurality of an ordered set of Ksource symbols, generating to a desired degree of accuracy one or moreof the not received source symbols among the ordered set of K sourcesymbols, wherein each symbol has an associated symbol relation that isdetermined by a systematic index, J(K), where J(K) is determined by K,wherein the value of each not received source symbol is determined bythe associated symbol relation and a plurality of L intermediate symbolvalues, wherein L is at least K, wherein the L intermediate symbolvalues are determined by the K source symbol values and by the K symbolrelations associated with the K source symbols and by a set of L-Kpre-coding relations, wherein the L intermediate symbol values can begenerated to a desired degree of accuracy from the N received source andrepair symbols, wherein the value of each not received source symbol isdetermined by the associated symbol relation and the plurality of Lintermediate symbol values.
 30. The method of claim 29, furthercomprising precalculating and storing the systematic index J(K) in atable for each relevant value of K.
 31. The method of claim 29, whereinthe systematic index J(K) for values of K between 4 and 8192 is asprovided in Appendix A.
 32. The method of claim 29, wherein the symbolrelations impose linear constraints on sets of symbols.
 33. The methodof claim 32, wherein the linear constraints are exclusive- orconstraints.
 34. The method of claim 29, wherein each symbol has anassociated ESI that identifies the symbol, wherein the systematic indexJ(K) and a value X, wherein X is a valid ESI, determines the symbolrelation for the symbol identified by ESI X.
 35. The method of claim 34,wherein the systematic index J(K) and the ESI X can be used to generatea triple (a[X],b[X],d[X]) that defines the symbol relation for thesymbol identified by ESI X.
 36. The method of claim 34, wherein eachsource symbol can be identified by an integer ESI in the range 0 to K−1,inclusive.
 37. The method of claim 34, wherein each repair symbol can beidentified by an integer ESI that is at least K.
 38. The method of claim35, wherein the triple (a[X],b[X],d[X]) that defines the source symbolrelation associated with the source symbol with ESI X places thefollowing linear constraint on the value C′[X] for source symbol withESI X and the values C[0], C[1], ..., C[L−1] of the L intermediatesymbols: defining an array of positive integer values B[0], ..., B[J−1],wherein   J is the minimum of d[X] and L, wherein L’ is the smallestinteger   that is prime that is at least the value of L, and wherein thevalues   of the array of positive integer values B[0],...,B[J−1]   areset using the method comprising:   setting the initial value of B[0] tob[X];   while B[0] is at least L, recalculating B[0] as the previous  value of B[0] plus a[X] modulo L’;   for values of j starting at 1 andgoing up to J−1 in increments of one,     initializing the value of B[j]to B[j−1] plus a[X];     while B[j] is at least L, recalculating B[j] asthe   previous value of B[j] plus a[X] modulo L’.


39. The method of claim 35, wherein the generation of the triple (a[X],b[X], d[X]) from an ESI X comprises the following steps:   calculating avalue A as (53591 + J(K) * 997) % 65521, wherein *     is themultiplication operator and % is the modulo operator;   calculating avalue B as 10267 * (J(K) + 1) % 65521;   calculating a value Y as (B +X * A) % 65521; and   determining a value v, wherein v is in the rangesfrom 0 and     1048575, inclusive, where v is calculated as the outputof a     random generator applied to the inputs Y, 0, and 1048575;


40. The method of claim 39 further comprising: determining d[X] based ona degree function applied to the input v.
 41. The method of claim 39further comprising: determining a[X], wherein a[X] is in the rangebetween 0 and L′−1, inclusive, wherein L′ is the smallest prime numbergreater than or equal to L, wherein L′ is calculated as one plus theoutput of the random generator applied to the inputs Y, 1 and L′−1. 42.The method of claim 39 further comprising: determining b[X], whereinb[X] is in the range between 0 and L′, inclusive, wherein b[X] iscalculated as the output of the random generator applied to the input Y,2 and L′.
 43. The method of claim 39, wherein the value of d[X] isdetermined based on v as follows: if the value of v is between 0 and10240, inclusive, then the value of d[X] is 1, if the value of v isbetween 10241 and 4911581, inclusive, then the value of d[X] is 2, ifthe value of v is between 4911582 and 712793, inclusive, then the valueof d[X] is 3, if the value of v is between 712794 and 831694, inclusive,then the value of d[X] is 4, if the value of v is between 831695 and948445, inclusive, then the value of d[X] is 10, if the value of v isbetween 948446 and 1032188, inclusive, then the value of d[X] is 11, ifthe value of v is between 1032189 and 1048575, inclusive, then the valueof d[X] is
 40.


44. The method of claim 39, wherein the random generator on inputs Y, i,and m is calculated as (V0[(Y+i) % 256]̂(V1[(floor(Y/256)+i) % 256]) % m,where % is the modulo operator, ̂ is the exclusive- or operator,/is thedivision operator, floor calculates the largest integer that is at mostthe input value, and V0 and V1 are each tables of 256 integers chosenrandomly or pseudo-randomly.
 45. The method of claim 44, wherein the 256values for table V0 is as described in Appendix B.1.
 46. The method ofclaim 44, wherein the 256 values for table V1 is as described inAppendix B.2.
 47. The method of claim 29, wherein the number L-K ofpre-coding relations comprises a first set of S pre-coding relations anda second set of H pre-coding relations, and wherein the L intermediatesymbols comprises a first set of K intermediate symbols, a second set ofS intermediate symbols, and a third set of H intermediate symbols. 48.The method of claim 47, wherein each pre-coding relation in the firstset of pre-coding relations is uniquely associated with an intermediatesymbol in the second set of intermediate symbols and is associated witha specified set of intermediate symbols among the first set ofintermediate symbols, wherein each pre-coding relation is a linearconstraint on the value of the associated intermediate symbol and thevalues of the intermediate symbols in the specified set.
 49. The methodof claim 48, wherein a linear constraint is an exclusive- or constraint.50. The method of claim 48, wherein for each first pre-coding relationamong the first set of pre-coding relations and for each secondpre-coding relation among the first set of pre-coding relations, thenumber of intermediate symbols that are both in the specified setassociated with the first pre-coding relation and in the specified setassociated with the second pre-coding relation is at most
 3. 51. Themethod of claim 48, wherein each of the intermediate symbols in thefirst set of intermediate symbols is in the specified set of exactlythree of the pre-coding relations in the first set of pre-codingrelations.
 52. The method of claim 47, wherein S is the smallest primeinteger that is at least as large as 0.01*K+X, wherein X is the smallestpositive integer such that X*(X−1) is at least 2*K, wherein H is thesmallest integer such that fact[H]/(fact[H−H′]*fact[H′]) is at leastK+2, wherein fact is the factorial operator, wherein H′ is the smallestinteger that is at least H/2.
 53. The method of claim 47, wherein eachpre-coding relation in the second set of pre-coding relations isuniquely associated with an intermediate symbol in the third set ofintermediate symbols and is associated with a specified set ofintermediate symbols among the first set and second set of intermediatesymbols, wherein the pre-coding relation is a linear constraint on theassociated intermediate symbol and the intermediate symbols in thespecified set.
 54. The method of claim 53, wherein a linear constraintis an exclusive- or constraint.
 55. The method of claim 53, wherein eachof the intermediate symbols in the first set or the second set ofintermediate symbols is in the specified set of approximately one-halfof the pre-coding relations in the second set of pre-coding relations.56. The method of claim 53, wherein for each first intermediate symbolin the first set or second set of intermediate symbols and for eachsecond intermediate symbol in the first set or second set ofintermediate symbols that is the next consecutive intermediate symbolfollowing the first intermediate symbol, the symmetric difference ofpre-coding relations in the second set of pre-coding relations of whichthe first intermediate symbol is a member of the specified set and ofwhich the second intermediate symbol is a member of the specified set isexactly two.
 57. The method of claim 1, wherein the symbols are placedinto one or more packets for transmission.
 58. The method of claim 29,wherein more than one source symbol is placed into at least one packet.59. The method of claim 6, wherein the symbols are placed into packetsfor transmission and wherein an ESI is included in each packet toidentify the first symbol placed into the packet.
 60. The method ofclaim 59, wherein more than one symbol is placed into at least onepacket and for packets that carry more than one symbol the ESIs for thesecond and subsequent symbols placed in the packet are determined by theESI for the first symbol placed in the packet.
 61. The method of claim60, wherein the ESI determined for the second symbol placed in a packetis one greater than the ESI placed in the packet that is associated withthe first symbol placed in the packet.
 62. The method of claim 34,wherein the symbols are received in one or more packets and wherein anESI is included in each packet to identify the first symbol placed intothe packet.
 63. The method of claim 62, wherein more than one symbol isplaced into at least one packet, wherein the ESIs for the second andsubsequent symbols placed in packets with more than one symbol aredetermined by the ESI for the first symbol placed in the packet.
 64. Themethod of claim 63, wherein the ESI determined for the second symbolplaced in a packet is one greater than the ESI placed in the packet thatis associated with the first symbol placed in the packet.
 65. The methodof claim 1 wherein the K source symbols correspond to a source block,wherein the source block is divided into N′ subblocks, wherein each ofthe N′ subblocks is composed of K′ sub-symbols, wherein one or moresource blocks can correspond to a source file, wherein each of the oneor more source blocks are encoded separately from the other sourceblocks.
 66. The method of claim 65 wherein the source file ispartitioned into one or more source blocks of approximately equal sizeas a function of the number of source symbols per source blocks and thenumber of source blocks, and wherein each source block is partitionedinto one or more subblocks of approximately equal size as a function ofthe size of a source symbol and the number of sublocks per source block.67. The method of claim 66 wherein more than one symbol is placed into apacket that is used for transmission.
 68. The method of claim 66 whereinone or more parameters used to partition the source file into one ormore source blocks and used to partition the one or more source blocksinto one or more subblocks are made known at the destination of thesymbols.
 69. The method of claim 1 wherein the K source symbolscorrespond to a source block, wherein the source block is defined by atransport protocol for streaming data.
 70. The method of claim 69wherein one or more parameters used to determine how the K sourcesymbols correspond to the source block are made known at the destinationof the symbols.
 71. The method of claim 69 wherein more than one symbolis placed into a packet that is used for transmission.
 72. The method ofclaim 29 wherein the K source symbols correspond to a source block,wherein the source block is defined by a transport protocol forstreaming data.
 73. The method of claim 72 wherein one or moreparameters used to determine how the K source symbols correspond to thesource block are made known at the destination of the symbols.
 74. Themethod of claim 72 wherein more than one symbol is placed into a packetthat is used for transmission.